The general solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=x^2$ is:

Solution

1. The given equation is $\frac{dy}{dx}+\frac{1}{x}y=x^2$, which is in the standard form $\frac{dy}{dx}+P(x)y=Q(x)$.
2. Calculate the integrating factor: $$\begin{array}{rl}I(x)& =e^{\int \frac{1}{x}dx}\\ & =e^{\ln |x|}\\ & =x\end{array}$$.
3. Multiply the entire differential equation by $x$ to get $x\frac{dy}{dx}+y=x^3$, which simplifies to $\frac{d}{dx}(xy)=x^3$.
4. Integrate both sides with respect to $x$ to obtain $$\begin{array}{rl}xy& =\int x^{3}dx\\ & =\frac{x^4}{4}+C\end{array}$$.
5. Divide by $x$ to isolate $y$, resulting in $y=\frac{x^3}{4}+\frac{C}{x}$.
6. Hence the answer is (B).