ODE4.3

$y''-y'+2y=10e^{{-x}}\sin(x)\,$

Find a particular solution $y_{p}\,$, by finding a generalization of the equation which has the original problem as the complex part. $y_{z}\,$ is the imaginary particular part.

$(D^{2}-D+2)y_{z}=10e^{{(-1+i)x}}\,$

The right hand side of the first equation is the complex part of the right hand side of the last equation. Solve for $y_{z}$ and find the imaginary part of the answer. This will be the answer to the original problem.

$y_{z}={\frac {10e^{{(-1+i)x}}}{(-1+i)^{2}-(-1+i)+2}}\,$

$y_{z}={\frac {10e^{{(-1+i)x}}}{3-3i}}={\frac {10}{3}}{\frac {(1+i)}{2}}e^{{-x}}(\cos(x)+i\sin(x))\,$

Find the imaginary part of the last equation.

$y_{p}={\mathrm {Im}}(y_{z})={\frac {5}{3}}e^{{-x}}(\cos(x)+\sin(x))\,$