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Math Help - inhomogenous eqn

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    inhomogenous eqn

    Does anyone know how to solve \frac{d^2y}{dx^2}+2\frac{dy}{dx}+2y=10exp(-x)sinx thnx
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    Quote Originally Posted by oxrigby View Post
    Does anyone know how to solve \frac{d^2y}{dx^2}+2\frac{dy}{dx}+2y=10exp(-x)sinx thnx

    First solve the homogenious equation. The auxillary equation is

    m^2+2m+2=0 \iff m^2+2m+1=-1 \iff (m+1)^2=-1 \iff m= -1 \pm i

    So the complimentry solution is

    y_c=c_1e^{-x}\cos(x)+c_2e^{-x}\sin(x)

    Since e^{-x}\sin(x) appears in the original equation the form of your particular solution is

    y_p=Axe^{-x}\cos(x)+Be^{-x}\sin(x)

    now take two derivatives to get

    y_p'=(A-Ax+Bx)e^{-x}\cos(x)+(-Ax+B-Bx)e^{-x}\sin(x)

    y_p''=(-2Ax+2B-2Bx)e^{-x}\cos(x)+(-2A+2Ax-2B)e^{-x}\sin(x)

    Now plug all of this into the ODE and simplify to get

    -2Ae^{-x}\sin(x)+2Be^{-x}\cos(x)=10e^{-x}\sin(x)

    So we can see that B=0,A=-5

    So y_p=-5xe^{-x}\cos(x)

    y=y_c+y_p
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