2015-05-08 09:05:55 -0500 | answered a question | solve differential equation Not sure if you still need an answer, but here goes; Define g(x) = f(x) - x*y, this makes the given equation homogeneous; d^2/dx^2 d/dy g(x, y) = g(x, y) And now assume g(x, y) = A(x)B(y). That lets you factorize the equation into two parts; d^2/xx^2 A(x) = a A(x), d/dy B(y) = (1/a) B(y), with some positive constant a. This can be solved as A(x) = C1 exp(+-sqrt(a) x), B(y) = C2 exp(y / a) And by multiplying them back, g(x, y) = (C1 * C2) exp(+-sqrt(a) x + y / a) = C exp(+-sqrt(a) x + y / a) Therefore, f(x, y) = C exp(+-sqrt(a) x + y / a) + x*y. General solution is linear combination of these solution with different values of C and a. But I might have missed something there, because this solution contains only two arbitrary parameters whereas the original equation is a third order differential equation. |

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