# What does an exact solution to a pde mean?

1. Nov 30, 2011

I've always heard the phrase "exact solution," but was never really sure what it meant. If I find a particular solution (not a general solution) to a PDE, is that solution considered an "exact solution"? (The solution satisfies given b.c. and i.c.)

2. Nov 30, 2011

### Pengwuino

An exact solution means that no approximation methods were used. MOST differential equations do not have exact, analytic solutions that you could write out as say, $f(x) = Ae^{kx^2} + Bx^2$; some nice simple function that works for all 'x'. Most equations require you to make assumptions about parts of the differential equation such as having to assume 'x' is much greater than some constant in the problem or that various values in the differential equation have certain realistic properties that make the equation solvable using approximate solutions. This is a quick answer as I don't have much time but there's other examples.

3. Dec 5, 2011

### bigfooted

A particular solution is an exact solution, but it is not the general solution.
Some people actually mean the general solution (or analytic solution) when they say exact solution.
u=0 is often an exact solution to a differential equation, but not a very interesting one and usually also not the general solution.