I am interested in solving the following pde $$a u v \partial_u f - (b + u v) \partial_v f = 0$$ over the reals, possibly with some restriction on the range of $a,b\in\mathbb{R}$ if necessary. I only need the characteristic curves, so there's no need to consider boundary conditions.
I was under the impression that all first order linear pdes were solvable algorithmically by the method of characteristics in terms of integrals of the coefficient functions in front of each derivative. However I gave this equation to Mathematica and it was not able to find a solution.
Did I misunderstand something here? Does this equation have some feature which means that it's not possible to solve it using characteristics? Is someone able to provide a solution to the equation?
$$a u v \partial_u f - (b + u v) \partial_v f = 0.$$ $f(u,v)$ is the unknown function.
Charpit-Lagrange : $$\frac{du}{auv}=\frac{dv}{-(b+uv)}$$ $$(b+uv(u))du+auv(u)dv=0$$ This is a non linear first order ODE very difficult to solve. With the invaluable help of WolframAlpha the solution is obtained on the form of an implicit equation : $$\sqrt{2\pi}\:\text{erf}\left(\frac{av+u}{\sqrt{2ab}}\right)+2i\frac{\sqrt{ab}}{u}\exp\left(-\frac{(av+u)^2}{2ab}\right)=c$$ The constant can be real or complex.
The general solution of the PDE expressed on implicit form $f(u,v)=\Phi(c)$ is : $$f(u,v)=\Phi\left(\sqrt{2\pi}\:\text{erf}\left(\frac{av+u}{\sqrt{2ab}}\right)+2i\frac{\sqrt{ab}}{u}\exp\left(-\frac{(av+u)^2}{2ab}\right)\right)$$ $\Phi$ is an arbitrary function (real or complex).
It is doubtfull that the function $\Phi$ can be analytically determined according to some boundary condition. In practice, better rely on numerical methods of solving.