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?