# Partial differential equation

• Jncik
In summary, the student attempted to solve the equation uxx = utt using the method of separation of variables. They found that X''(x) - λ*X(x) = 0 and T''(t) - λ*Τ(t) = 0. They found that for λ = 0, they got X(x) = Ax + B where A and B are constants. When λ > 0, they got u(x,t) = http://img690.imageshack.us/img690/8068/asdqu.gif [Broken] [Broken][/URL] where c1,c2,C1,C2 constants. Finally, they

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

Last edited by a moderator:
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.)

Note that given any function f, f(x-t) or f(x+t) satisfies this equation.

Are you given any boundary conditions?

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