= Solution
{c}
The second-order <differential operator> factors as
$$
\boxed{xv_{xx}+(2y+\beta x)v_{xy}=\bigl(x\partial_x+(2y+\beta x)\partial_y\bigr)(v_x).}
$$
Therefore setting $u=v_x$ turns the second-order <partial differential equation> into exactly the first-order <partial differential equation> in part (a). Conversely, given a sufficiently smooth <solution of a differential equation> $u$ of that equation on a coordinate rectangle, define
$$
v(x,y)=\int_{x_*}^{x}u(\xi,y)\,d\xi+F(y).
$$
The <fundamental theorem of calculus> gives $v_x=u$, so this solves the second-order equation, with an arbitrary twice continuously differentiable <function> $F$. This classical correspondence is local on smooth patches; for the cornered solution of part (b), it can instead be interpreted using <distributional derivatives>.
Back to article page
