For the initial value problem y' −xy = x, y(0) = 1. write down the first four nonzero terms of the series

Not clear if you're supposed to find a series solution to the ODE, or solve exactly then give the requested terms of the series expansion of the that solution...

The exact solution is easy enough to find. We can separate the variables:





Given that , we have



Then recall that



to write the solution as




so the first four terms of the series are

The answer is option c.