Step 1

From Example 1 in Section 14.2, the corresponding homogeneous equation has the particular solution

x = e^{-sin t}.

Step 2

The function v(t) is expressed by an integral.

We cannot evaluate the integral, so we leave it in this form. It does not matter which value is chosen for the lower endpoint in the integral, so we take the lower endpoint zero.

Step 3

The general solution is y - vx + Cx,

or