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Mar 15, 2019
EarMaster Pro 6: An individualized method to improve music theory training.Q:
solution of a symbolic PDE?
$u_x+u_y=f(u)$
How can i solve this? what kind of method/techniques i should use?
A:
The equation you have is what's called a partial differential equation. In general, you would use the following method of attack:
Separate variables. This means rewriting the equation as an ordinary differential equation: $ rac{dx}{du} = rac{dy}{du} = rac{du}{df(u)}$.
Change dependent variable. In your example, $u = x+y$ and $x+y = z$.
Solve the resulting first order linear ODE. (You can use anything that can integrate, from a pencil and paper to a computer algebra system, and from there you can solve the implicit equation $x+y=z$ for $y$ in terms of $x$.)
Integrate the solution of step 2 to get $f(u(x,y))$.
After each step, you should re-check your calculations.
In general, the first step is to choose a particular substitution for $u(x,y)$. Substitutions that simplify the equation often make the problem easy, or at least simpler. In your example, the substitution $u(x,y) = x+y$ is one that simplifies the PDE, but it also creates new problems. For example, you may lose information about the original function $f(u)$. Also, in general, there may be multiple functions satisfying the above description. You may need to make some additional assumptions, and check them to see if your solution is unique.
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