Diff. eq.

Thread Starter

boks

Joined Oct 10, 2008
218
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...
 
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blazedaces

Joined Jul 24, 2008
130
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
 
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