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1. A PDE could be very general, linear and nonlinear;
2. The saparable solution could also be general. Suppose your PDE is about u with only two variables, x and y. Then the solution can be written as u(x,y) and, therefore, the saparable solution could be u(x,y)=X(x)Y or X(x)+Y or maybe some other forms.
3. It also depends on what you want to achieve. For example, if you only want to have a taste of the solution of the equation (du/dx)^2 + (du/dy)^2=1, you may want to try u(x,y)=X(x)+Y and see what kind of solutions you can get.
4. Do you have any initial or boundary conditions?
In mathematical physics, the linear PDE's of the second order play a very important role. There are two nice things about them:
1. they are linear, so that the superposition of the solutions prevails. Therfore, you can build up any linear combinations of the solutions to come up with a new solution.
2. They have the order 2. This is imprtant, because most of the eigenvalue problems, which people are familiar with, are of the second order. When you sum up those eigen solutions (which look trivial), you end up with a nontrivial solution.
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