Answer to Question #122433 in Differential Equations for junior

Question

d^2y/dx^2 + 3(dy/dx) + 2y = sin(e^x )

Answer 1

$k^2+3k+2=0$

$k_1=-2$

$k_2=-1$

$y(x)=C_1e^{k_1x}+C_2e^{k_2x}$

$y(x)=C_1e^{-2x}+C_2e^{-x}$

$e^{-x}\frac {d}{dx}C_2(x)+e^{-2x}\frac {d}{dx}C_1(x)=0$

$\frac {d}{dx}(e^{-x})\frac {d}{dx}C_2(x)+\frac {d}{dx}(e^{-2x})\frac {d}{dx}C_1(x)=sin(e^x)$

$e^{-x}\frac {d}{dx}C_2(x)+e^{-2x}\frac {d}{dx}C_1(x)=0$

$-e^{-x}\frac {d}{dx}C_2(x)-2e^{-2x}\frac {d}{dx}C_1(x)=sin(e^x)$

$\frac {d}{dx}C_1(x)=-e^{2x}sin(e^x)$

$\frac {d}{dx}C_2(x)=e^{x}sin(e^x)$

$C_1(x)=-\int e^{2x}sin(e^x)dx=e^xcos(e^x)-sin(e^x)+C_3$

$C_2(x)=\int e^{x}sin(e^x)dx=C_4-cos(e^x)$

Answer:

$y(x)=C_3e^{-2x}+C_4e^{-x}-e^{-2x}sin(e^x)$