Tricomi equation

Remarks

The equation is elliptic where $y > 0$, hyperbolic where $y < 0$, and parabolic on the line $y = 0$, so it is of mixed type. In the half-plane $y < 0$ its characteristic curves are $3x \pm 2(-y)^{3/2} = \text{const}$. Some authors write it as $u_{xx} = x \, u_{yy}$, which is the same equation after renaming and reflecting the variables.

Example: $u(x, y) = x^2 - \tfrac{1}{3} y^3$ is a solution, because $y \, u_{xx} + u_{yy} = 2y - 2y = 0$.

Non-example: $u(x, y) = x^2 + y^2$ is not a solution, because $y \, u_{xx} + u_{yy} = 2y + 2$ is not identically $0$.