Differential Equations

Differential Equations By Pui chor Wong September 18, 2004 DeVry Calgary for Math230

Introduction • Differntial equation is an equation that contains a derivative or a differential. • Example: y'=3x2+8x-6 • Example: xdy=4xy+y2dx

Order • The order of a differential equation is the highest derivative in the equation • Example: y'+x3-x=4 is called a first order differential equation • Example: xy'+y2y''=8 is called a second order differential equation

Degree • The degree of a differential equation is the power of the highest order derivative • Example xy''+y2y'-3y=6 is called the first degree second order differential equation • Example (y''')2+3y'=0 is a second-degree, third-order differential equation

Solution • A solution to a differential equation is a relationship between the variable and differentials that satisfes the equation • In general, a differential equation has an infinite family of solutions. That is called general solution. • The solution of an nth-order differential equation can have at most n arbitrary constants. • A solution having the maximum number of constants is called the general solution or complete solution. • When additional information is given to determine at least one of the conditions, the solution is then called particular solution

Separation of Variables • One kind of DE (differential equation) can be solved by separating the variables and integrate. • A first-order, first degree differential equation y'=f(x,y) is called separable if it can be written in the form • y'=A(x)/B(x) or A(x)dx=B(y)y

Integrable Combinations • Consider the product rule or quotient rule • d(xy) = xdy+ydx • d(x/y) = xdy-ydx/x2

Linear Differential Equation • A general procedure to solve a first order linear differential equation • A first-order differential equation is aid to be linear if it can be written in the form • y' + P(x)y = Q(x)

Solution using integrating factor

Second and Higher Order • Direct Integration by reduction of the order if possible • Example y''=A(x) • y'=A(x)dx • repeat until y appears on the left side of the equation • Linear or nonlinear, homogeneous or non-homgeneous, too complicated • focus on linear, constant coefficients higher order differential equation

Homogeneous Equations • If Q(x)=0, it is called homogeneous • If Q(x)!=0, it is called nonhomogeneous • Rewrite equation using D operator • y'' should be written as D2y • y' should be written as Dy • D opertor is not an algebraic quantity but can be treated as so.

General solution • (D - p1)y1=0 means y1= C1 e p1x • (D - p2)y2=0 means y2= C2 e p2x • (D - p1)(D - p2)y=0 means y= C1 e p1x+C2 e p2x

Distinct, Repeated, Complex • Distinct: refer to previous case • Repeated: y = C1e-px + C2xe-px • Complex: y = e-ax(C1cos(bx) + C2sin(bx))

Non-Homogeneous • y = yc + yp • yc is called complementary solution • yp is called particular solution • use previous method to find complementary solution by letting Q(x)=0 first • The to find particular solution, choose if Q(x) is of the form xn, yp will be of the form A+Bx+Cx2+..+kxn • Solving for undetermined coefficients A, B,.. C etc.

Particular solution.. • If Q(x) is of the form aebx, yp is of the form Aebx • If Q(x) is of the form axebx, yp is of the form Aebx+Bxebx • If Q(x) is of the form a cos(bx) and a sin(bx), then yp is of the form Asin(bx)+Bcos(bx) • If Q(x) is of the form axcos(bx) or axsin(bx), then yp is of the form Asin(bx)+Bcos(bx)+Cxcos(bx)+Exsin(bx)

Differential Equations