# Diff. eq.

1. Nov 27, 2008

### leopard

1. The problem statement, all variables and given/known data

The function u(x,t) satisfies the equation

(1) $$u_{xx}$$ = $$u_{tt}$$ for 0 < x < pi, t > 0

and the boundary conditions

(2) $$u_x$$(0,t) = $$u_x$$(pi, t) = 0

Show that (1) and (2) satisfy the superposition principle.

2. The attempt at a solution

I let w(x,t) = au(x,t) + bv(x,t) for two constants a and b.

$$w_{tt}$$ = $$au_{tt}$$ + $$bv_{tt}$$ = $$au_{xx}$$ + $$bv_{xx}$$ = $$cw_{xx}$$, where c is a constant

Have I now showed that w(x,t) satisfies (1)? $$w_{xx}$$ is not equal to $$w_{tt}$$ unless c is 1...

Last edited: Nov 27, 2008
2. Nov 27, 2008

### HallsofIvy

Staff Emeritus
You said you let w= au+ bv. What is c? What do you mean by "auxx+ bvxx= c wxx"? I don't see where that comes from.

Perhaps it would be simpler to see if you rewrote the equation as uxx- utt= 0. What is wxx- wtt?