# Diff. eq.

#### 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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#### boks

Joined Oct 10, 2008
218
I figured it out, thanks.

I'm studying PDEs.

#### blazedaces

Joined Jul 24, 2008
130
I figured it out, thanks.

I'm studying PDEs.
I see. That's good.

Cheers,

-blazed