Given the pde
eqn:=0.3846153846*(diff(F(x, y), y, y))+diff(F(x, y), x)-(diff(w(x), x, x))*y-(1/2)*(int(diff(F(x, y), x), y = -1 .. 1))
+diff(F(x, y), x, x)-(diff(w(x), x, x, x))*y-(1/2)*(int(diff(F(x, y), x, x), y = -1 .. 1));
will produce the "solution"
sol:= F(x, y) = (diff(w(x), x))*y-_C1*y/exp(x)-(1923076923/5000000000)*_c*x*y+_C2*y+
In this expression _C1, _c, _C2, and _C4 are arbitrary numeric constants: not functions of anything. There are a couple of "interesting" aspects to this equation:
- The existence of the sub-expression _C2*y+_C4*y: since both _C2 and _C4 are arbitrary and numeric, then they can be combined into _C3*y: why didn't Maple do this - don't really know
- The existence of the "indexed" arbitrary constant _c: why is this an "indexed" constant? - particularly given that there is no _c or _c. I haven't seen an indexed constant returned by a call to pdsolve() before: and I have no idea why it is occurring in this case
You should laso be aware that the last term _F1(x) represents an arbitrary function of 'x'. preceding terms in the solution are functions of 'y', (or 'x' and 'y'), but the existence of _F1(x), means that any function of 'x' alone *ought* to be OK
Tidying the given solution, means that
sol:=F(x, y) = (diff(w(x), x))*y-_C1*y/exp(x)-(1923076923/5000000000)*_C2*x*y+_C3*y+
ought to be a valid solution: and in fact checking this with
confirms, that the "tidied" version of the solution is still a solution.
I have laso tried the tidied version when substituting various functions for _F1(x), and pdetest() confrims that each of these is a solution (NB, I only tried functions which were continous, and differentiable)
Overall a very interesting PDE - idle curiousity: does it arise as a result of some "physical, real world" type of problem, or is more or less an "exercise in mathematics"?