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?

Multipying both side by $e^{-2\ln x}=1/x^2$ we get  $$y'=\frac{2}{x}y+x^3\Longleftrightarrow (\frac{y}{x^2})'= x\Longleftrightarrow \frac{y}{x^2}=x^2/2+c\\\Longleftrightarrow y(x)= x^4/2+cx^2$$
