finding particular solution to this pde

May 2014
8
0
Texas
\(\displaystyle U_{yy} - x^2u = x^2\)

I found the general solution to be
\(\displaystyle U(x,y) = f(x)e^{-xy} + g(x)e^{xy} + A + A1x + A2x^2\)

How do I find the constants to the particular solution for a pde?
 
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skipjack

Forum Staff
Dec 2006
21,481
2,470
Didn't you mean U instead of u? In that case, U = -1 satisfies the equation.
 

Country Boy

Math Team
Jan 2015
3,261
899
Alabama
This is NOT, by the way, a partial differential equation. Although U depends upon 2 variables, x and y, the only derivative is with respect to y so this is an ordinary differential equation with x as a parameter.

As skipjack said, the constant function, U= -1 obviously satisfies the equation \(\displaystyle U_{yy}- x^2U= x^2\).
 

skipjack

Forum Staff
Dec 2006
21,481
2,470
I understand that point of view, but the "parameter" is allowed to vary whilst y varies, as a partial derivative was specified. It's still technically a PDE, which is why $f(x)$ and $g(x)$ appear in the solution.

The $A1x$ and $A2x^2$ terms should have used $y$ instead of $x$, but the terms should have been omitted anyway as there were no terms in $y$ on the right-hand side of the original equation.

Although $A1$ and $A2$ turn out to be zero and $A$ turns out to be -1, any of these could be non-constant functions of $x$ for this kind of equation, so they shouldn't be referred to as constants.