Question: How do I solve a partial differential equation for f(x(t),y(t),t)?

Hello,

I’m just toying around with diff(), dsolve(), and, as it happens, also with pdsolve(). Let’s say I have an equation as the following:

DF := (diff(f(x, y, t), `$`(x, 2)))*omega+(diff(f(x, y, t), `$`(y, 2)))^2+diff(f(x(t), y(t), t), t) = 5;

A function f, depending on x, y and t where x and y also depend on t. I multiply the second partial derivative of f by x with ω, add it to the square of the second partial derivative of f by y and add the total derivative by t, which should give me 5.

If I input this into pdsolve(), I get the following:

pdsolve(DF, f);
Error, (in pdsolve/info) the indeterminate function named f is expected to depend on the same variables throughout the equation but came as {f(x, y, t), f(x(t), y(t), t)}

So it basically complains that one time f is f(x,y,t) and the other time, it’s f(x(t),y(t),t). But in order to get the total derivative, I have to write it as f(x(t),y(t),t), and to get the partial derivative, I have to use the other version. Something like »diff(f(x(t),y(t),t),x(t))« (or x(t)$2) doesn’t work either.

My other approach is to do the total derivation by hand:

DE := (diff(f(x, y, t), `$`(x, 2)))*omega+(diff(f(x, y, t), `$`(y, 2)))^2+(diff(f(x, y, t), x))*(diff(x(t), t))+(diff(f(x, y, t), y))*(diff(y(t), t))+diff(f(x, y, t), t) = 5

Unfortunately, Maple also does not like this one:

pdsolve(DE, f);
Error, (in pdsolve/sys/info) ambiguous input: the variables {x, y} and the functions {f(x, y, t), x(t), y(t)} cannot both appear in the system

Is my input that senseless that there is no meaningful representation, or didn’t I just find it yet?