Mastering PDEs: Solving the Non-Constant Coefficient d^2G/dxdy Equation

[toptrial]

d^2G/dxdy+(a-1)*dG/dx*dG/dy=0
where G is a function of x and y.

Moreover, what if a is not a constant, but instead a function of x and y?

 

[NoMoreExams]

Can't you assume a solution of the form G(x,y) = X(x)Y(y)

 

[toptrial]

I am afraid not.
In fact, I know G(x,y)=-ln([1+(a-1){X(x)+Y(y)}]^[1/(a-1)]) is the solution. I am just trying to figure out how to slove this pde.
Thanks

 

[MathematicalPhysicist]

If you assume G(x,y)=X(x)+Y(y)
will that do the trick?

You'll get:
(a-1)XY=0, then either X=0 or Y=0 or a=1, the last one is uninterseting, so you have two solutions here.

There not much to go out here, you use multiplication or addition, division and substraction are defined apostriori by them.

Edit: or any other composition of the elementary functions.

I am quite sure the way your textbook or teacher made this question, that they knew already the answer, and then found which equation it satisifies, you know the solution so take the derivative and see his process of devising the question.

 

[RedBranchKnight]

For some problems like this one can sometimes use the transformation

z = x + y
w = x - y

Maybe some terms drop off and a new simpler PDE is achieved? It's like a rotation.

 

[Big-T]

What RedBranchKnight refers to is a special case of the http://en.wikipedia.org/wiki/Method_of_characteristics" .

 

[Redbelly98]

If you assume G(x,y)=X(x)+Y(y)
will that do the trick?

You'll get:
(a-1)XY=0, then either X=0 or Y=0 or a=1, the last one is uninterseting, so you have two solutions here.

Um, aren't all of those uninteresting cases? This is saying that any G which is a function of just one of the variables (x or y, but not both) is a solution. In that case, one of the 1st-derivatives will be zero, as will the mixed 2nd-derivative. The original equation becomes 0=0.

 

[MathematicalPhysicist]

Well, I understand now, the solution's of toptrial is the answer itself, because you don't know what is X(x), Y(y), you need to take the derivative of G(x,y), and find what are X(x) and Y(y).
Or so I think.

 

[RedBranchKnight]

This pde looks like a Goursat problem

http://mathworld.wolfram.com/GoursatProblem.html

 

[toptrial]

Hi guys, how would I solve this pde numerically in matlab?

d^2G/dxdy+A*dG/dx*dG/dy=0
where G and A are both functions of x and y.