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...

 

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...

You have not shown that w(x,t) satisfies (1). May I ask what "\(w_{xx}\) is not equal to \(w_{tt}\) unless c is 1..." have to do with the superposition principle?

The part that is going about it correctly is when you do the following: "\(au_{tt}\) + \(bv_{tt}\) = \(au_{xx}\) + \(bv_{xx}\)". But you need to prove that to be true, you can't just write it down...

By the way, are you studying waves at the moment, because part 1 is part of the definition of a wave...

-blazed

 

I figured it out, thanks.

I'm studying PDEs.

 

I figured it out, thanks.

I'm studying PDEs.

I see. That's good.

Cheers,

-blazed