Superposition Principle for u(x,t) in Diff. Eq.

[leopard]

Homework Statement

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...

 

[HallsofIvy]

You said you let w= au+ bv. What is c? What do you mean by "au_xx+ bv_xx= c w_xx"? I don't see where that comes from.

Perhaps it would be simpler to see if you rewrote the equation as u_xx- u_tt= 0. What is w_xx- w_tt?