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Tutorial 9

Tutorial 9. Second Order Linear Differential Equations, part II. Homogeneous Linear Equations with Constant Coefficients. Consider a nonhomogeneous equation with constant coefficients:

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Tutorial 9

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  1. Tutorial 9 Second Order Linear Differential Equations, part II Tutorial 9

  2. Homogeneous Linear Equations with Constant Coefficients Consider a nonhomogeneous equation with constant coefficients: where a, b and c are constants and g(x) is an exponent, polynom or harmonic function (ewx; a0xn+…an; sin(wx) or cos(wx) ) or their product. This type of equation is very useful in many applied problems (physics, electrical engineering, etc..). Then one should make an intelligent guess about the form of the solution, up to the constant multipliers, and then substitute this guess into the equation to calculate the values of the multipliers. Tutorial 9

  3. Example 1 Consider a nonhomogeneous equation with constant coefficients: Suppose, the solution is y=A·sin(x)+B·cos(x), then: y`=A·cos(x)-B·sin(x) y``=-A·sin(x)-B·cos(x) Substituting, we obtain: (-A-3B-4A)cos(x)+(-B+3A-4B)sin(x)=2sin(x) From where we obtain -5A-3B=0 3A-5B=2 A=3/17;B=-5/17; Y=1/17(3cos(x)-5sin(y)) Tutorial 9

  4. Example 2 Consider a nonhomogeneous equation with constant coefficients: Suppose, the solution is y=Ax2, then: y`=2Ax y``=2A Substituting, we obtain: 2A-6Ax-4Ax2=4x2 We see that there are no solutions in Ax2. Now, try Ax2+Bx+C. And obtain y=-x2+3/2x-13/8. Tutorial 9

  5. Example 3 Consider a nonhomogeneous equation with constant coefficients: Two linearly independent solutions of the homogeneous equation are y1=cos(x) and y2=sin(x) For a particular solution yp=u1cos(x)+u2sin(x) Then yp=[-u1sin(x)+u2cos(x)]+[…=0] Differentiating again, and substituting: u1’(x)cos(x)+u2’(x)sin(x)=0; -u1’(x)sin(x)+u2’(x)cos(x)=sec(x). Solving, we obtain: Tutorial 9

  6. Example 3 (Solution) u1’(x)=-tan(x); u2’(x)=1; u1(x)=ln(cos(x)); u2(x)=x; Therefore, the particular solution is yp(x)=xsinx+cos(x)ln(cos(x)) And the general solution is y=c1cos(x)+c2sin(x)+xsin(x)+cos(x)ln(cos(x)) Tutorial 9

  7. Example 4 Solve the system: Solution: Assuming that x=aert, we obtain the system of algebraic equations, whose determinant is: Tutorial 9

  8. Example 4. Solution. Therefore r1=1, r2=2, r3=-1. The corresponding eigenvectors are: The general solution is Tutorial 9

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