Isolate equation with two variables

I'd like to separate two unknown symbolic equations for their variables.

For example using separable partial differential equations to solve the laplace equation:

syms f(x) g(y) u(x,y);
u=f(x)*g(y);

eqn=a*diff(u,2,x)+c*diff(u,2,y)==0;
isolate(eqn,x)

I want to use this for unknown user input and transform it to something like

-c*g(y)/diff(g(y), y, y)==a*f(x)/diff(f(x), x, x)

however, the isolate function only puts all arguments of y or x on one side as its not possible to solve for them leaving the equation in form of

g(y)*diff(f(x), x, x) + f(x)*diff(g(y), y, y) == 0

a and c are users inputs, however there is more input and laplace is only an example so that I can't just separate it manually.

Is there any way to get the equations in the form I need?

Thanks in advance

Answer 1

[arushi]

Hi Arne,

To separate the variables in a partial differential equation like the Laplace equation, you can use the method of separation of variables. The goal is to rewrite the equation in a form where one side depends only on ( x ) and the other side depends only on ( y ). This is possible when the equation is homogeneous and linear.

Here’s how you can implement this in MATLAB :

syms f(x) g(y) lambda a c
eqn1 = diff(f,2) == -lambda*(c/a)*f;
eqn2 = diff(g,2) == lambda*g;

% Now you can solve eqn1 and eqn2 for f(x) and g(y) respectively

Hope this helps.