Originally Posted by
spearfish Hey Guys, I am stuck on the following problem:
Show that U(x,t) = e^(lambda^2*alpha^2*t)*[A*sin(lambda*x) + B*cos(lambda*x) satisfied the PDE Ut = alpha^2 * Uxx for arbitrary A,B and lambda.
Here's what I have so far, but I don't know if it's right:
1.) General eqn to soln: U(x,t) = Summation * Bn * Rn * Tn
2.) (R*T)t = alpha^2 *(RT)xx
(R*T')/RT = (alpha^2 * Rxx * T)/RT
T/T' = (alpha^2*R'')/R = -lambda^2 (just some constant)
T' = - lambda^2*T = T(t) = e^(-lambda^2*t)
R'' = (-lambda^2 * R) / (alpha^2)
R(x) = A*cos((lambda*x)/alpha) + B*sin((lambda*x)/alpha)
3.) Plug in R(x) and T(x) into general solution
U(x,t) = e^-lambda^2*t * (A*cos((lambda*x)/alpha) + B*sin
((lambda*x)/alpha))
I know that this is not the final solution, and maybe it's even the wrong approach, but I don't know what to do next. Any and all help would be greatly appreciated. Thanks for the help!