Question

Solve the initial value problem y"-7y'+6y=-e^(x)(17cosx-7sinx), y(0)=4, y'(0)=2.

Answer 1

you assume the particular solution is a linear combination of the right-hand side and all of its linearly independent derivatives.

Answer 2

ignoring the constants
d/dx e^x cos x = e^x cos x - e^x sin x
and similarly for
d dx e^x sin x = e^x sin x + e^x cos x

we see that there are two independent terms, and we assume the solution yp is
yp= A e^x cos x + B e^x cos x