Find the particular solution for (D^2 + 3D + 2)y = sin e^x

Question

find y_c and y_p

Answer 1

The above equation is factored to (D+1)(D+2)y = sin(e^x)

Then if (D+2)y = z, then the top equation becomes (D+1)z = sin(ex)

When you solve for z, you find that z = -e^-xcos(e^x) + C₁e^-x

Where C₁ = the first constant of integration

Going back to the first equation, we now know that (D+2)y = -e^-xcos(e^x) + C₁e^-x

Solving this for y, we find that y = -e^-2xsin(e^x) + C₁e^-x + C₂e^-2x