We will work through Leibniz' technique, using

$x\frac{dy}{dx}+y=3x^2$

as an example.

$($a$)$ Rewrite Equation $(4)$ in the form that Leibniz used to begin his process. ${?}^6$ What are the functions $m$ and $n?$

$($b$)$ Leibniz then defined a new function $p$ that satisfies the condition

$\frac{dp}{p}=ndx$

Using $n$ from part $($a$),$ solve for $p.$

$($c$)$ With these functions, use Leibniz' method to verify that

$\frac{{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}}{b}ar(mpdx)+py=0$

solves the original differential equation for the form of thr equation from part $($a$).$

$($d$)$ At this point we know $m,n,$ and $p$ so the only unknown is $y.$ Solve for $y$ and show it solves Equation $(4).$