We are asked to solve the differential equation $xy' + y = x^3$.

1. Problem Description

We are asked to solve the differential equation $xy' + y = x^3$.

2. Solution Steps

The given differential equation is $xy' + y = x^3$. We can rewrite this equation as

$\frac{d}{dx}(xy) = x^3$

Integrating both sides with respect to $x$, we have

$\int \frac{d}{dx}(xy) dx = \int x^3 dx$

$xy = \frac{x^4}{4} + C$

Dividing both sides by $x$, we get

$y = \frac{x^3}{4} + \frac{C}{x}$

3. Final Answer

The solution to the differential equation is $y = \frac{x^3}{4} + \frac{C}{x}$.