non homo. 2nd order diff.eqn

2009-05-26 4:21 am

using the solution of the non homogeneous second order differential equation:
ay''+by'+cy=f(x), y=yh+yp

solve: y''+4y'+29y=52e^(-x) + (841x)

the answer is y= e^(-2x) (Acos(5x)+Bsin(5x))+ 2e^(-x) + 29x - 4

please show the answer steps by steps, thanks!

回答 (1)

Prof. Physics

2009-05-26 4:38 am

✔ 最佳答案

y" + 4y' + 29y = 52e-x + 841x

Auxiliary equation: k2 + 4k + 29 = 0

k = -2 +- 5i

Complementary solution, yh

= e-2x(Acos5x + Bsin5x)

By the superposition principle, let particular solution, yp

= Ce-x + Dx + E

yp' = -Ce-x + D

yp" = Ce-x

Putting back to the equation,

Ce-x + 4(-Ce-x + D) + 29(Ce-x + Dx + E)= 52e-x + 841x

26Ce-x + 29Dx + 4D + 29E = 52e-x + 841x

Comparing coefficients, C = 2, D = 29, E = -4

So, the general solution: y

= yh + yp

= e-2x(Acos5x + Bsin5x) + 2e-x + 29x - 4

參考: Physics king

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