# DSolve does not solve 2nd order linear PDE with variable coefficients

I find that DSolve can solve

DSolve[x*D[u[x, y], {x, 2}] - D[u[x, y], x, y] == 0, u, {x, y}] 

or

DSolve[x*D[u[x, y], {x, 2}] - x *D[u[x, y], x, y] == 0, u, {x, y}]

or

DSolve[x*D[u[x, y], {x, 2}] - 3*D[u[x, y], x, y] == 0, u, {x, y}]}

but cannot solve

DSolve[x*D[u[x, y], {x, 2}] - a *D[u[x, y], x, y] == 0, u, {x, y}]

or

DSolve[x*D[u[x, y], {x, 2}] - y *D[u[x, y], x, y] == 0, u, {x, y}]

Can I re-pose the equation to make it work? thanks, GB

MMa 10.3, 11.1

• DSolve[]'s support for PDE equations is still somewhat limited, so don't be surprised if some things don't work yet.See Results:12000.org/my_notes/pde_in_CAS/maple_2019_and_mma_12/index.htm – Mariusz Iwaniuk Jul 10 at 21:14
• You can break the last PDE down into two successive quadratures: Fold[ Function[{eq, var}, DSolve[eq /. First[#1], var, {x, y}]] @@ #2 &, {{}}, Transpose@{{x*D[v[x, y], {x}] - y*D[v[x, y], y] == 0, D[u[x, y], x] == v[x, y]}, {v, u}}] – Michael E2 Jul 11 at 5:15
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• @MichaelE2 Yes, something comes out. Being thick between the ears, I have not been successful in backtesting. It is possible to do so? – Gary Bollenbach Jul 13 at 17:31
• @MichaelE2. Now it seems to check. Thanks for passing on this useful technique. – Gary Bollenbach Jul 14 at 22:10

\$Assumptions = a > 0

• Assumptions -> a \[Element] Reals && a != 0 also works. – Michael E2 Jul 11 at 0:01