Solving the Differential equation: $y'=\frac{2}{x}y+x^3$ We have the differential equation $$y'=\frac{2}{x}y+x^3$$ and we know $x \in (0, \infty)$.
My attempt with variation of constants
\begin{align}
\phi(x) &= \exp \left(\int \frac{2}{x} dx \right) \\
&= \exp(2\ln|x|) \\
&= x^2c
\end{align}
and 
\begin{align}
\psi(x) &= (x^2c) \cdot \int \frac{x^3}{x^2} dx \\
&= (x^2c) \cdot \frac{x^2}{2}
\end{align}
but this solution is wrong. Where is the mistake?

Observe that
$$\phi\left(x\right)=\exp\left(\int\frac{2}{x}{\rm d}x\right)=\exp\left(2\ln\left|x\right|+C\right)=Ax^{2}$$
with $A\equiv\exp C$. Thus the particular solution is
$$y_{\rm p}=Ax^{2}\int\frac{x^{3}}{Ax^{2}}{\rm d}x=\frac{x^{4}}{2}$$
