Q.Solve $\dfrac{dy}{dx}+\dfrac{2}{x}y=x^2$ (for $x>0$).

Identifying $P=2/x$ gives an integrating factor of $x^2$; multiplying through and integrating gives the general solution.

Identifying $P$ and $Q$

Comparing with $\frac{dy}{dx}+Py=Q$: here $P=\frac{2}{x}$, $Q=x^2$.

Finding the integrating factor

Applying the standard solution formula

Check (differentiate the solution back into the ORIGINAL equation): $\frac{dy}{dx}=\frac{3x^2}{5}-\frac{2C}{x^3}$. Then …

Computing $e^{\int(2/x)dx}$ incorrectly as $2\ln x$ (forgetting to exponentiate) instead of the correct $e^{2\ln x}=x^2$ — the integrating factor is a …