For nonzero constant , prove that

is a particular solution of the equation

Find the general solution of the equation

*Proof.* The general solution of the homogeneous equation is . Then, particular solutions of the homogeneous equation are and (taking and , respectively). The Wronskian of and is then

So, we have the functions (of Theorem 8.9) and given by

Therefore we have a particular solution of the non-homogeneous equation given by

Using this theorem, the general solution of

is

(where we changed the values of the constants and in the last step to absorb the extra and terms).

At first I thought “this is nonsense! the integral clearly has a zero derivative and so it cannot be a solution” but then after some pondering I noticed that the integral is not an indefinite integral since the integrand is not fixed (it contains an x).