Partial differential equation

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  • #1
Jncik
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Homework Statement



solve uxx = utt

The Attempt at a Solution



using the method of separation of variables I get

X''(x) - λ*X(x) = 0
T''(t) - λ*Τ(t) = 0

For λ = 0 I get X(x) = Ax + B where A and B are constants
T(t) = Dt + E where D and E are constants

hence u(x,t) = (Ax + B)*(Dt + E)

for λ>0 I get u(x,t) = [PLAIN]http://img690.imageshack.us/img690/8068/asdqu.gif [Broken] [Broken][/URL] where c1,c2,C1,C2 constants

for λ = -b^2<0 I get (E*cos(bx) + F*sin(bx))*(G*cos(bt) + H*sin(bt)) where E,F,G,H constants

hence the solution

is from the superposition principle

u(x,t) = (Ax + B)*(Dt + E) + [PLAIN]http://img690.imageshack.us/img690/8068/asdqu.gif [Broken] [Broken][/URL] + (E*cos(bx) + F*sin(bx))*(G*cos(bt) + H*sin(bt))


please is this correct? i think its wrong, because I am not sure about the last part

also our professor has only the [PLAIN]http://img690.imageshack.us/img690/8068/asdqu.gif [Broken] [Broken][/URL] as a reply, but i think its wrong, because we have to check for all λ to find all the possible solutions
 
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Answers and Replies

  • #2
grey_earl
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λ is fixed. You either have λ > 0 or λ < 0, so you have either the solution
with exp[√(λ) x] or the one with the sines and cosines. Apart from that, you are correct but for one (I assume) typo: it's not exp[√(λx)] but exp[√(λ) x].

(PS: Of course you could also have λ = 0, but that appears seldom in reality.)
 
  • #3
phyzguy
Science Advisor
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Note that given any function f, f(x-t) or f(x+t) satisfies this equation.
 
  • #4
hunt_mat
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Are you given any boundary conditions?
 
  • #5
Jncik
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thanks for your answers, no there are no boundary conditions

there is another exercise with boundary conditions and I get a simple result..
 
  • #6
HallsofIvy
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Without boundary or initial conditions, the most general possible solution is Af(x-t)+ Bf(x+t) where f is any twice differentiable function of a single variable, A and B constants, as phyzguy said.

That can be written in the form Jncik gives initially by summing over all possible values of [itex]\lambda[/itex].
 

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