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Cengage Maths Solutions Class 12 Differential Equations - Calculus

Cengage Maths Solutions Class 12 Differential Equations PDfs provides accurate solutions to practice before your final exam. Students can study this comprehensive Cengage Maths PDf Solutions on Differential Equations and its various sub-topics for effective exam preparation. The Cengage 12 PDF solutions help the student to swiftly practice and revise the important concepts just before the annual exam to score high marks in maths.<br>For more such video solutions/Pdf solution, visit https://doubtnut.com and clear your doubt instantly.

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Cengage Maths Solutions Class 12 Differential Equations - Calculus

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  1. CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER DIFFERENTIAL EQUATIONS || CALCULUS  Download Doubtnut Today Ques No. Question CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential 1 dy Find the order and degree of the following differential equation: 1 + y = dy dx dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential Find d3y the order and degree of the following differential equation: d2y 2 dx3 − x + y = 0 e dx2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential Find the order and degree of the following differential equation: 3 dy sin−1( ) = x + y dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential Find the order and degree of the following differential equation: 4 dy ln( ) = ax + by dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential Find the order and degree of the following differential equation: 1

  2. 5 1 6 d2y 4 dy = [y + ( ] ) dx2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential 5 4 d2y 3 dy 6 = {1 + ( } Find the order and degree (if defined) of the equation: ) dx2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential d3y dy 7 Find the order and degree (if defined) of the equation: = xln( ) dx3 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential Find d4y the order and degree (if defined) of the equation: 6 d2y 8 + 3( ) + sinx = 2cosx dx2 dx4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential Find the order and degree (if defined) of the equation: 2

  3. 9 2 d3y d2y 3 dy ( ) + 4 − 3 + 5 = 0 dx3 dx2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential 3 2] 2 dy 1[1 + ( ) dx 10 Find the order and degree (if defined) of the equation: a = , d2y dx2 where is constant a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential d4y d3y 11 Find the order and degree (if defined) of the equation: − sin( ) = 0 dx3 dx4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of 12 Form the differential equation of family of lines concurrent at the origin.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of Form the differential equation of all concentric circles at the origin. 13  Watch Free Video Solution on Doubtnut

  4. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of Form the differential equation of all circle touching the x-axis at the origin and centre on the y-axis. 14  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of Form the differential equation of the family of parabolas with focus at the origin and the axis of symmetry along the axis. 15  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of Form the differential equation of family of lines situated at a constant distance from the origin. p 16  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of 17 Find the differential equation of all parabolas whose axis are parallel to the x-axis.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of 2 tanx a − b If then show that {tan− 1( ) } y = a + b 2 √a2− b2 18 2

  5. d2y bsinx . = dx2 (a + bcosx)2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of 19 Find the differential equation of all non-vertical lines in a plane.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of Find the differential equation of all the ellipses whose center is at origin and axis are co-ordinate axis. 20  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of y2 x2 Consider the equation where a and b are specified + = 1, 21 a2+ λ b2+ λ constants and is an arbitrary parameter. Find a differential equation satisfied by it. λ  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of Find the degree of the differential equation satisfying the relation √1 + x2+ √1 + y2 22 = λ(x√1 + y2− y√1 + x2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of Find (1 + e2x)dy + (1 + y2)exdx = 0, the particular solution of differential when y = 1 the equation 23 given that x = 0.  Watch Free Video Solution on Doubtnut

  6. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of log(dy) 24 Solve , given that when = 4x − 2y − 2 y = 1 x = 1. dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of dy 25 Solve given that when = x + 1, x = 0,y = 3. e dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of 1 + y2 dy 26 Solve the differential equation (1 + x + x2) = xy 1 + x2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of dy 27 = (x + y)2 Solve dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of dy Solve 28 √1 + x + y = x + y − 1 dx  Watch Free Video Solution on Doubtnut

  7. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of 2 d2y dy 29 Solve = ( ) dx2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of x x 30 Solve ye dx = (xe + y2)dy(y ≠ 0). y y  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of 31 Solve the equation: sec2xtanydx + sec2ytanxdy = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of y f( ) xdy = (y + x )dx x Solve 32 y f′( ) x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of m =

  8. Find the real value of for which the substitution m dy will transform the y = um 33 differential equation in to a homogeneous equation. 2x4y + y4= 4x6 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of siny + x dy 34 Solve = sin2y − xcosy dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of x + 2y + 3 dy 35 Solve = 2x + 3y + 4 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of x − 2y + 5 dy 36 Solve = 2x + y − 1 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous Find the sum ∑ ∑ jnCi 37 0≤i<j≤n−1  Watch Free Video Solution on Doubtnut

  9. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous Solve [2√xy − x]dy + ydx = 0 38  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous dy 39 Solve x( ) = y(logy − logx + 1) dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous y π 40 Solve when [xsin2( ) − y]dx + xdy = 0; y = x = 1. 4 x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous Show that the differential equation homogeneous equation. can be reduced to a y3dy + (x + y2)dx = 0 41  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous 2x − y + 1 dy 42 Solve = x + 2y − 3 dx

  10.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous dy 43 Solve x2( ) + y = 1 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous e−2√x y dx [ ] Solve 44 − = 1(x ≠ 0) dy √x √x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous 45 Solve ydx − xdy + logxdx = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous (x − etan^ ( − 1)y)dy 46 Solve (1 + y2) + = 0 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous What is dx the integrating factor of the differential equation 47 y (1 − y2) + = ay( − 1 < y < 1) dy x  Watch Free Video Solution on Doubtnut

  11. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous dy 48 Solve the equation + ycotx = sinx dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous dy 49 Solve the equation (x + y + 1)( ) = 1 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous dy 50 Solve the equation (1 − x2)( ) + 2xy = x√1 − x2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear dy y 51 Solve the equation = 2y1ny + y − x dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear Solve the equation ydx + (x − y2)dy = 0 52  Watch Free Video Solution on Doubtnut

  12. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear Find the equation of a curve passing through −(y + y3) and having gradient (0,1) 53 at(x,y) 1 + x + xy2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear e1 Solve (xy2− dx − x2ydy = 0 ) 54 x3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear dy y 55 Solve ) = y3 ( ) + ( dx x

  13.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear dy 56 Solve ) = ex−y(ex− ey). ( dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear 1 57 Solve (x − 1)dy + ydx = x(x − 1)y dx. 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear ey 1 dy 58 Solve the equation + = x2 dx x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear dy 59 Solve the equation + xsin2y = x3cos2y dx  Watch Free Video Solution on Doubtnut

  14. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear dy xy Solve the equation 60 + = x√y (1 − x2) dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear dy 61 Solve the equation = (x3− 2xtan− 1y)(1 + y2) dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear 62 Find the orthogonal trajectory of (a being the parameter). y2= 4ax  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear 63 Find the orthogonal trajectories of xy = ⋅  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear Find the equation of the curve in which the subnormal varies as the square of the ordinate. 64  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear 65 Find the curve for which the length of normal is equal to the radius vector.  Watch Free Video Solution on Doubtnut

  15. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear Find the curve for which the perpendicular from the foot of the ordinate to the tangent is of constant length. 66  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear A curve lines are drawn parallel to the co-ordinate axes. If the curve divides the area formed by these lines and co-ordinates axes in the ratio passes through the origin. Through any point y = f(x) on the curve, (x,y) 67 find the curve. m:n,  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear 68 Find the orthogonal trajectories of family of curves x2+ y2= cx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear 69 Find the orthogonal trajectory of (a being the parameter). y2= 4ax  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear What constant interest rate is required if an initial deposit placed into an account accrues interest compounded continuously is to double its value in six years? (ln|x| = 0. 6930) 70  Watch Free Video Solution on Doubtnut

  16. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form OfCENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation Find the time required for a cylindrical tank of radius and height a round hole of area at the bottom. The flow through the hole is according to the law a to empty through r H , where and , are respectively, the velocity of flow h(t) v(t) = k√2gh(t) v(t) 71 through the hole and the height of the water level above the hole at time the acceleration due to gravity. and is t, g  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation Suppose that a mothball loses volume by evaporation at a rate proportional to its instantaneous area. If the diameter of the ball decreases from 2cm to 1cm in 3 months, how long will it take until the ball has practically gone? 72  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation A body at a temperature of . If the rate of change of the temperature of a body is proportional to the temperature difference between the body and its surrounding medium. If after 5 min the temperature of the body is , find (a) how long it will take the body to reach a temperature of 75 F and (b) the temperature of the body after 20 min. ^ 0 is placed outdoors where the temperature is 500F 1000F 73 600F  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation Find the time required for a cylindrical tank of radius 2.5 m and height 3 m to empty through a round hole of 2.5 cm with a velocity water in the tank. m/s, being the depth of the h 74 2.5√h  Watch Free Video Solution on Doubtnut

  17. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form OfCENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation If the population of country double in 50 years, in how many years will it triple under the assumption that the rate of increase is proportional to the number of inhabitants. 75  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation The rate at which a substance cools in moving air is proportional to the difference between the temperatures of the substance and that of the air. If the temperature of the air is 290 K and the substance cools from 370 K to 330 K in 10 min, when will the temperature be 295 K? 76  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation A normal is drawn at a point of a curve. It meets the x-axis and the y-axis in P(x,y) 1 1 77 point AND A respectively, such that =1, where is the origin. Find B, + O OA OB the equation of such a curve passing through (5, 4)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation dy x + y y4 dx Solve = x2+ 2y2+ 78 x2 dy y − x dx  Watch Free Video Solution on Doubtnut

  18. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form OfCENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation Find the equation of the curve which is such that the area of the rectangle constructed on the abscissa of any point and the intercept of the tangent at this point on the y-axis is equal to 4. 79  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation Solve y y {xcos( ) + ysin( )}ydx x x 80 y y = {ysin( ) − xcos( )}xdy x x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation Find the equation of the curve passing through the origin if the middle point of the segment of its normal from any point of the curve to the x-axis lies on the parabola 2y2= x 81  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation (x + y)2 dy 82 Solve = (x + 2)(y − 2) dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_General Form Of Variable Separation 2 dy dy Solve given that y( ) 83 + 2x − y = 0 y(0) = √5 dx dx  Watch Free Video Solution on Doubtnut

  19. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical d 84 If y + (xy) = x(sinx + logx), f ∈ dy(x). dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical x 85 If then find ty(t)dt = x2+ y(x), ∫ y(x) a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical Given a function ' g' which has a derivative g'(0) = 2 and g(x + y) = eyg(x) + eyg(y) for every real x and satisfies for all x and y then: g'(x) 86  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical Let the function In is defined where exists for is fixed f(x) f(x) x ≥ 2 and k d 87 positive real numbers prove that if then (x. f(x)) ≥ − kf(x) dx where A is independent of x. f(x) ≥ Ax−1−k  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical

  20. dy If and are the solution of the differential equation , where 88 + Py = Q y1 y2 P dx Q z = 1 + ⋅ e−f dx, and where is an arbitrary constant. c are functions of alone and , then prove that y2= y1z y1 Q x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical dy If and are two solutions to the differential equation . + P(x)y = Q(x) y1 y2 89 is the general solution to the equation where dx Then prove that is any constant. y = y1+ c(y1− y2) c  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical Find a pair of curves such that (a) the tangents drawn at points with equal abscissas intersect on the y-axis. (b) the normal drawn at points with equal abscissas intersect on the x-axis. (c) one curve passes through (1,1) and other passes through (2, 3). 90  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical Given two curves: passing through the point and y = f(x) (0, 1) x 1 passing through the point f(t)dt The tangents drawn to (0, ). g(x) = ∫ n − ∞ 91 both the curves at the points with equal abscissas intersect on the x-axis. Find the curve y = f(x).  Watch Free Video Solution on Doubtnut

  21. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical A cyclist moving on a level road at 4 m/s stops pedalling and lets the wheels come to rest. The retardation of the cycle has two components: a constant 0.08 friction in the working parts and a resistance of meters per second. What distance is traversed by the cycle before it comes to rest? (consider 1n 5=1.61). due to m/s2 92 , where is speed in v 0.02v2/m  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The force of resistance encountered by water on a motor boat of mass water with velocity is proportional to the velocity At then engine shuts off. Find an expression for the position of motor boat at time and also the distance travelled by the boat before it comes to rest. Take the proportionality constant as k > 0. going in still when its velocity is m t = 0 v0, v v. 93 t  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The degree of the differential equation satisfying √1 − x2+ √1 − y2= a(x − y) 94 is (a) (b) (c) (d) none of these 1 2 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The differential equation whose solution is arbitrary constants, is of (a) second order and second degree (b) first order and second degree (c) first order and first degree (d) second order and first degree , where and A are Ax2+ By2= 1 B 95  Watch Free Video Solution on Doubtnut

  22. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The differential equation of the family of curves and are arbitrary constants is A B where y = ex(Acosx + Bsinx), 96  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical 97 Differential equation of the family of circles touching the line at is y = 2 (0,2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical 98 The differential equation of all parabolas whose axis are parallel to the y-axis is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical A differential equation associated to the primitive is derivative w.r.t. yn nth (a)y3+ 2y2− y1= 0(b)4y3+ 5y2− 20y1 is (where y = a + be5x+ ce−7x x) 99 = 0(c)y3+ 2y2− 35y1= 0 (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The order and degree of the differential equation of all tangent lines to the parabola is (a) 1,2 (b) 2,3 (c) 2,1 (d) 1,1 y = x2 100  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Statistical 101 The form of the differential equation of the central conics is ax2+ by2= 1  Watch Free Video Solution on Doubtnut

  23. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Statistical The differential equation for the family of curve arbitrary constant, is where is an x2+ y2− 2ay = 0, a 102  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Statistical dy 103 = (ey− x)−1, If where , then is expressed explicitly as y y(0) = 0 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Physical x If (where is an arbitrary constant) is the general solution of the c y = log|cx| 104 dy y x x differential equation then the function is + φ( ), φ( ) = y y dx x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Physical The y = c1cos(x + c2) − c3e( −x+c4)+ (c5sinx), differential equation whose general solution where is given by are 105 c1,c2,c3, c4, c5 arbitrary constants, is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Physical ′= 0, log + (1) = ,

  24. 106 The solution to the differential equation where is ylogy + xy′= 0, y(1) = e,  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Physical 2 + sinx dy π 107 If and then equals ( ) = − cosx, y(0) = 1, y( ) y = y(x) y + 1 2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Physical y − 1 The equation of the curves through the point (1, 0) and whose slope is is 108 x2+ x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Physical x(2logx + 1) dy 109 The solution of the equation is = siny + ycosy dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Physical dy 110 The solution of the equation log is ( ) = ax + by dx  Watch Free Video Solution on Doubtnut

  25. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Physical The solution of the equation is given by (x2y + x2)dx + y2(x − 1)dy = 0 111  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation A normal is drawn at a point of a curve. It meets the x-axis and the y-axis in P(x,y) 1 1 112 point AND A respectively, such that =1, where is the origin. Find B, + O OA OB the equation of such a curve passing through (5, 4)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation dy x + y y4 dx Solve = x2+ 2y2+ 113 x2 dy y − x dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous Find the equation of the curve which is such that the area of the rectangle constructed on the abscissa of any point and the intercept of the tangent at this point on the y-axis is equal to 4. 114  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous

  26. Solve y y {xcos( ) + ysin( )}ydx 115 x x y y = {ysin( ) − xcos( )}xdy x x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation Find the equation of the curve passing through the origin if the middle point of the segment of its normal from any point of the curve to the x-axis lies on the parabola 2y2= x 116  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation (x + y)2 dy 117 Solve = (x + 2)(y − 2) dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation 2 dy dy Solve given that y( ) + 2x − y = 0 y(0) = √5 118 dx dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Different Case Of Bounded Area d 119 If y + (xy) = x(sinx + logx), f ∈ dy(x). dx  Watch Free Video Solution on Doubtnut

  27. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation x 120 If then find ty(t)dt = x2+ y(x), ∫ y(x) a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Statistical Given a function ' g' which has a derivative g'(0) = 2 and g(x + y) = eyg(x) + eyg(y) for every real x and satisfies for all x and y then: g'(x) 121  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear Let the function In is defined where exists for is fixed f(x) f(x) x ≥ 2 and k d 122 positive real numbers prove that if then (x. f(x)) ≥ − kf(x) dx where A is independent of x. f(x) ≥ Ax−1−k  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear dy If and are the solution of the differential equation , where + Py = Q y1 y2 P dx 123 Q z = 1 + ⋅ e−f dx, and where is an arbitrary constant. c are functions of alone and , then prove that y2= y1z y1 Q x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear

  28. dy If and are two solutions to the differential equation . 124 + P(x)y = Q(x) y1 y2 is the general solution to the equation where dx Then prove that is any constant. y = y1+ c(y1− y2) c  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical Find a pair of curves such that (a) the tangents drawn at points with equal abscissas intersect on the y-axis. (b) the normal drawn at points with equal abscissas intersect on the x-axis. (c) one curve passes through (1,1) and other passes through (2, 3). 125  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical Given two curves: passing through the point and y = f(x) (0, 1) x 1 passing through the point f(t)dt The tangents drawn to (0, ). g(x) = ∫ 126 n − ∞ both the curves at the points with equal abscissas intersect on the x-axis. Find the curve y = f(x).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical A cyclist moving on a level road at 4 m/s stops pedalling and lets the wheels come to rest. The retardation of the cycle has two components: a constant 0.08 friction in the working parts and a resistance of meters per second. What distance is traversed by the cycle before it comes to rest? (consider 1n 5=1.61). due to m/s2 127 , where is speed in v 0.02v2/m  Watch Free Video Solution on Doubtnut

  29. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Statistical The force of resistance encountered by water on a motor boat of mass water with velocity is proportional to the velocity At then engine shuts off. Find an expression for the position of motor boat at time and also the distance travelled by the boat before it comes to rest. Take the proportionality constant as k > 0. going in still when its velocity is m t = 0 v0, v v. 128 t  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Statistical The degree of the differential equation satisfying √1 − x2+ √1 − y2= a(x − y) 129 is (a) (b) (c) (d) none of these 1 2 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential The differential equation whose solution is arbitrary constants, is of (a) second order and second degree (b) first order and second degree (c) first order and first degree (d) second order and first degree , where and A are Ax2+ By2= 1 B 130  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential The differential equation of the family of curves and are arbitrary constants is A B where y = ex(Acosx + Bsinx), 131  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential = 2 (0,2)

  30. 132 Differential equation of the family of circles touching the line at is y = 2 (0,2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of 133 The differential equation of all parabolas whose axis are parallel to the y-axis is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of A differential equation associated to the primitive is derivative w.r.t. yn nth (a)y3+ 2y2− y1= 0(b)4y3+ 5y2− 20y1 is (where y = a + be5x+ ce−7x x) 134 = 0(c)y3+ 2y2− 35y1= 0 (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of The order and degree of the differential equation of all tangent lines to the parabola is (a) 1,2 (b) 2,3 (c) 2,1 (d) 1,1 y = x2 135  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential 136 The form of the differential equation of the central conics is ax2+ by2= 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of The differential equation for the family of curve arbitrary constant, is where is an x2+ y2− 2ay = 0, a 137  Watch Free Video Solution on Doubtnut

  31. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of dy 138 = (ey− x)−1, If where , then is expressed explicitly as y y(0) = 0 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation x If (where is an arbitrary constant) is the general solution of the c y = log|cx| 139 dy y x x differential equation then the function is + φ( ), φ( ) = y y dx x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of The y = c1cos(x + c2) − c3e( −x+c4)+ (c5sinx), differential equation whose general solution where is given by are 140 c1,c2,c3, c4, c5 arbitrary constants, is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation 141 The solution to the differential equation where is ylogy + xy′= 0, y(1) = e,  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation 2 + sin

  32. 142 2 + sinx dy π If and then equals ( ) = − cosx, y(0) = 1, y( ) y = y(x) y + 1 2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation y − 1 143 The equation of the curves through the point (1, 0) and whose slope is is x2+ x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of x(2logx + 1) dy The solution of the equation is 144 = siny + ycosy dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of dy 145 The solution of the equation log is ( ) = ax + by dx

  33.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation The solution of the equation is given by (x2y + x2)dx + y2(x − 1)dy = 0 146  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear 147 Solution of differential equation is dy − sinxsinydx = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of dv k 148 The solution of is + v = − g dt m  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear dy 149 The solution of the equation is = cos(x − y) dx  Watch Free Video Solution on Doubtnut

  34. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation dy 150 Solution of is + 2xy = y dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Different Case Of Bounded Area The general x + y solution of the differential equation 151 x − y dy is + sin( ) = sin( ) 2 2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of 152 The solution of is (x + y + 1)dy = dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous x2dy y 153 The solution of is − xy = 1 + cos( ) dx x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation π The slope of the tangent at to a curve passing through (x,y) is given by ) (1, 154 4 y y then the equation of the curve is − cos2( ), x x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous dy If then the solution of the equation is 155 = y(logy − logx + 1), x dx  Watch Free Video Solution on Doubtnut

  35. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation y2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ f( ) y2 x2 The solution of differential equation is 156 yy′= x + y2 x2 f′( ) x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous The solution of is (x2+ xy)dy = (x2+ y2)dx 157  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation 158 The solution of is (y + x + 5)dy = (y − x + 1)dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The slope of the tangent at x2+ y2 to a curve passing through a point (x,y) is (2,1) 159 , then the equation of the curve is 2xy  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation Solution of the differential equation

  36. 160 (y + x√xy(x + y))dx + (y√xy(x + y) − x)dy = 0 is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear The general solution of the differential equation, where is a known function, is ϕ(x) , y′+ yϕ′(x) − ϕ(x)ϕ′(x) = 0 161  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation x2+ y2+ 1 dy 162 The solution of satisfying is given by = y(1) = 1 2xy dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear dy The integrating factor of the differential equation is (x(log)ex) + y = 2(log)ex 163 dx given by  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear

  37. 164 dy The solution of the differential equation x(x2+ 1)( ) = y(1 − x2) + x3logx dx is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear dy 165 Integrating factor of differential equation is cosx + ysinx = 1 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear Solution of the equation dy π cos2x − (tan2x)y = cos4x,|x| < , 4 dx 166 3√3 π wheny( ) = is 6 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear If integrating factor of is equal to is then x(1 − x2)dy + (2x2y − y − ax3)dx = 0 e∫pdx, P 167  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear = ( )

  38. A function satisfies y = f(x) (ex)2 (x + 1)f′(x) − 2(x2+ x)f(x) = 168 , (x + 1) ∀x ≻ 1. f(0) = 5, If then is f(x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation 1 dy The solution of the differential equation is 169 = xy[x2siny2+ 1] dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear dy 170 The general solution of the equation is = 1 + xy dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation (x + 2y3)dy 171 The solution of the differential equation is = y dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation 1 1 dy The solution of the differential equation x2 cos( ) − ysin( ) = − 1, 172 dx x x where as is y → − 1 x → ∞  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation dy The solution of the differential equation given 2x2y = tan(x2y2) − 2xy2, 173 dx π is y(1) = , 2  Watch Free Video Solution on Doubtnut

  39. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear Solution of the differential equation y2 x2 1 1 { }dx + { }dy − − 174 (x − y)2 (x − y)2 y x is = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation φ(xy) dy If then is equal to 175 y + x = x φ(xy) φ′(xy) dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation The solution of differential equation is (2y + xy3)dx + (x + x2y2)dy = 0 176  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation x ye−dx − (xe( − )+ y3)dy = 0 x 177 The solution of is y y

  40.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation y(x + y3) dy The curve satisfying the equation and passing through the point = 178 x(y3− x) dx is (4, − 2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous 3x2y4+ 2xy dy The solution of the differential equation is = 179 x2− 2x3y3 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation The solution of the differential equation {1 + x√(x2+ y2)}dx 180 + {√(x2+ y2) − 1}ydy = 0 is equal to  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation Which of the following is not the differential equation of family of curves whose tangent π

  41. π form an angle of with the hyperbola ? xy = c2 4 181 x − y dy dy dy x (a) = (b) = (c) x + y x − y dx dx dx x + y = x − y (d)N.O.T.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical Tangent to a curve intercepts the y-axis at a point tangent through passes through another point (1,0). The differential equation of the curve is P A line perpendicular to this P. 182  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The differential equation of the curve for which the initial ordinate of any tangent is equal to the corresponding subnormal (a) is linear (b) is homogeneous of second degree (c) has separable variables (d) is of second order 183  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical 2 = a( ) 2 2 Orthogonal trajectories of family of the curve arbitrary constant, is , where is any 184 3 x + y 3 a 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical 185 The differential equation of all non-horizontal lines in a plane is  Watch Free Video Solution on Doubtnut

  42. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The curve in the first quadrant for which the normal at any point joining the origin to that point form an isosceles triangle with the x-axis as base is (a) an ellipse (b) a rectangular hyperbola (c) a circle (d) None of these and the line (x,y) 186  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The equation of the curve which is such that the portion of the axis of between the origin and tangent at any point is proportional to the ordinate of that point is cut off x 187  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical A normal at on a curve meets the x-axis at x(1 + y2) and is the foot of the P(x,y) Q N 188 ordinate at If , then the equation of curve given that it passes NQ = P. 1 + x2 through the point is (3, 1)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical A curve is such that the mid-point of the portion of the tangent intercepted between the point where the tangent is drawn and the point where the tangent meets the y-axis lies on the line If the curve passes through y = x. 189 then the curve is (1, 0),  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical

  43. 7 1 The equation of a curve passing through and having gradient at (2, ) 190 1 − x2 2 is (x, y)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical A normal at any point the axis, the differential equation of the curve is to the curve (x,y) cuts a triangle of unit area with y = f(x) 191  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The normal to a curve at origin is twice the abscissa of hyperbola (d) ellipse meet the x-axis at , then the curve is a (a) parabola (b) circle (c) P If the distance of from the P(x, y) G. G 192  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical if are two positive numbers such that a, b f(a + x) = b + [b3+ 1 − 3b2f(x) 193 1 + 3b{f(x)}2− {f(x)}3] 3 for all real , then prove that x is peroidic and find its peroid? f(x)

  44.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The equation of a curve passing through (1,0) for which the product of the abscissa of a point and the intercept made by a normal at square of the radius vector of the point is P 194 on the x-axis equal twice the P P  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The differential equation of all parabolas each of which has a latus rectum whose axes are parallel to the x-axis is (a) of order 1 and degree 2 (b) of order 2 and degree 3 (c) of order 2 and degree 1 (d) of order 2 and degree 2 and 4a 195  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical The curve with the property that the projection of the ordinate on the normal is constant and has a length equal to is a 196  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical y(2x4+ y)dy 197 The solution of differential equation is given by = (1 − 4xy2)x2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A

  45. Differential Equation The solution of the differential equation (xcoty + logcosx)dy 198 + (logsiny − ytanx)dx = 0 is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Physical spherical rain drop evaporates at a rate proportional to its surface area. The differential equation corresponding to the rate of change of the radius of the rain drop if the constant of proportionality is is K > 0 199  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Statistical Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops is proportional to the square root of water depth where the constant of proportionality on the acceleration due to gravity and the geometry of the hole. If is measured in 1 depends y, k > 0 200 t minutes and then the time to drain the tank if the water is 4 m deep to start k = , with is (a) 30 min (b) 45 min (c) 60 min (d) 80 min 15  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Statistical The population of a country increases at a rate proportional to the number of inhabitants. is the population which doubles in 30 years, then the population will triple in approximately. (a) 30 years (b) 45 years (c) 48 years (d) 54 years 201 f  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Statistical An object falling from rest in air is subject not only to the gravitational force but also to air resistance. Assume that the air resistance is proportional to the velocity with constant of proportionality as , and acts in a direction opposite to motion k > 0 202 m Then velocity cannot exceed. (g = 9.8 ). s2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous The solution of differential equation x2= 1 − 2( 2 ( ) )

  46. − 2( 2 dy x ( ) y ) −1 dy x dx +( ) + 203 2! y dx −3( 3 dy x ( ) y ) dx + + ...is 3!  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation 2 204 The solution of the differential equation is y′y' ' '= 3(y' ')  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation Number of values of d3y for which is a solution of the differential y = emx m ∈ N d2y 205 dy equation (a) 0 (b) 1 (c) 2 (d) More than 2 − 3 − 4 + 12y = 0 dx3 dx2 dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Geometrical A curve passing through x ty(t)dt = x2y(x),(x > 0) and satisfying the differential equation (2,3) 206 is ∫ 0

  47.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation d2y The solution of the differential equation when = sin3x + ex+ x2 y1(0) = 1 207 dx2 and is y(0)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Variable Separation EQUATIONS_Method Of x3 x5 x + + + dx − dy 3! 5! 208 The solution of the differential equation = is x2 x4 dx + dy 1 + + + 2! 4!  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation The solution of the differential equation 2 x2y2 dy dy ( ) x = 1 + xy + 2! dx dx 209 3 x3y3 dy ( ) + + ...is 3! dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of y x The differential equation of the curve is 210 + = 1 c − 1 c + 1

  48.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of θ d dx 211 The function satisfies the differential equation ∫ f(θ) = 1 − cosθcosx d(θ) 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Formation Of A Differential equation of the family of curves where and are B v = + B, 212 A r arbitrary constants, is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation The solution 1 of the differential equation where y' ' '− 8y' '= 0, 213 , is , y′(0) = 0,y0= 1 y(0) = 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation dy 214 (ex2+ ey2)y + ex2(xy2− x) = 0is The solution of the differential equation dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous Which one of the following function(s) is/are homogeneous? x − y (a)f(x, y) = (b)f(x,y) x2+ y2 x 1 2 = x y− tan−1( 215 )(c)f(x, y) 3 3 y x = x(ln√x2+ y2− lny) + ye y (d) none of these  Watch Free Video Solution on Doubtnut

  49. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations Of First Order And Degree EQUATIONS_Differential (x − h)2+ (y − k)2= a2(a For the differential equation whose solution is constant), its (a) order is 2 (b) order is 3 (c) degree is 2 (d) degree is 3 is a 216  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation The equation 2 of the curve satisfying the differential equation dy dy 217 can be a (a) circle (b) Straight line (c) Parabola − x = 0 y( ) + (x − y) dx dx (d) Ellipse  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Differential Equations EQUATIONS_Linear Which of the following equation(s) is/are linear? dy y dy = lnx(b)y( ) + 4x (a) + dx x dx 218 dy = 0(c)(2x + y3)( ) = 3y(d)N. O. T. dx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation ax + h dy The solution of represent a parabola when (a) (a) (b) = a = 0, b ≠ 0 219 by + k dx (c) (d) a ≠ 0,b ≠ 0 b = 0, a ≠ 0 a = 0,b ∈ R  Watch Free Video Solution on Doubtnut

  50. CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Applications Of Differential Equation EQUATIONS_Statistical The equation of the curve satisfying the differential equation passing through the point (0,1) and having slope of tangent at and represent 2nd and 1st order derivative), then (a) increasing function (b) three distinct real roots (d) y = f(x) y2(x2+ 1) = 2xy1 as 3 (where is a strictly y = f(x) x = 0 y2 220 y1 is a non-monotonic function (c) has only one negative root. has a y = f(x) y = f(x))  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL Equations EQUATIONS_Homogeneous tany y Identify the statement(s) which is/are true. (a) is a f(x,y) = e + x x y2 lny y homogeneous of degree zero. (b) is a ⋅ sin−1( )dy = 0 dx + x x f(x, y) = x2+ sinxcosy x x 221 homogeneous differential equation. (c) homogeneous. (d) differential equation. is a non homogeneous is a (x2+ y2)dx − (xy2− y3)dy = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_DIFFERENTIAL EQUATIONS_Solution Of A Differential Equation The graph of the function passing through the point (0,1) and satisfying the y = f(x) dy differential equation is such that (a) it is a constant function. + ycosx = cosx 222 dx (b) it is periodic (c) it is neither an even nor an odd function. (d) it is continuous and differentiable for all f(x)  Watch Free Video Solution on Doubtnut

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