Q.Solve the differential equation $x\dfrac{dy}{dx} + y = x^2$, $x>0$.

A first-order equation linear in $y$ has the standard form $\dfrac{dy}{dx}+Py=Q$ ($P,Q$ functions of $x$, or constants); multiplying it through by the integrating factor $\text{I.F.}=e^{\int P\,dx}$ turns the left-hand side into the exact derivative $\dfrac{d}{dx}(y\cdot\text{I.F.})$, so that integrating both sides directly gives the general solution $y\cdot\text{I.F.}=\displaystyle\int Q\cdot\text{I.F.}\,dx+C$. When an equation is instead linear in $x$ as a function of $y$ -- standard form $\dfrac{dx}{dy}+Px=Q$, $P,Q$ functions of $y$ -- the identical reasoning with $x$ and $y$ exchanged gives $\text{I.F.}=e^{\int P\,dy}$ and general solution $x\cdot\text{I.F.}=\displaystyle\int Q\cdot\text{I.F.}\,dy+C$; which of the two forms applies is decided by inspecting which variable the equation is actually linear in.