Find the particular solution of the differential equation

satisfying the initial condition and when .

This is a second-order linear differential equation of the form

These values of and give us . So, and . By Theorem 8.7 (pages 326-327 of Apostol) we then have

We then use the initial conditions and to solve for and , letting and ,

Then, we can simplify these expressions for and ,

And

Therefore, we have

Shouldn’t the -1 be inside a paranthesis with x not a and b?