PDE - Boundary & Initial Conditions Problem

[Mathrican-American]

I have a problem!
u_t = u_xx, 0 < x < 4, t > 0
with boundary conditions u(0, t) = 0, u(4, t) = 0
and initial condition u(x, 0) = f(x)


I used the method of separation of variables to start out:
Let u(x, t) = X(x)*T(t)
X(x)*T'(t) = X''(x)*T(t) (Our original equation)
T'(t) = X''(x) = λ (Divide both sides of the equation by X(x)*T(t))
T(t) X(x)
Set each side of this equation equal to some constant, λ
We now have two equations:
X''(x) = λ*X(x)
T'(t) = λ*T(t)

When we apply the boundary conditions, we find:
u(0, t) = X(0)*T(t) = 0
We know that T(t) ≠ 0, so X(0) must = 0
u(4, t) = X(4)*T(t) = 0
X(4) must also = 0

I just have little-to-no idea of where to go from here. We weren't assigned a textbook for this class, so I only have my professor's notes to go off of, which are pretty hard to follow.
Could anyone be kind enough to explain to me what I'm attempting to do here & steer me in the right direction as to how I finish this problem? Any help at all would be much appreciated.
Thank you!

 

[HallsofIvy]

Generally speaking, people do not take "Partial Differential Equations" until after they have studied "Ordinary Differential Equations". Are you saying you have never taken any class that dealt with ordinary differential equations, whether it was called that or not?

Certainly before you would be expected to deal with equations like this, you should know how to solve "ordinary linear equations with constant coefficients", about the simplest possible kind of differential equation. If you have a differential equation of the form ay''+ by'+ cy= 0, you can form the associated "characteristic equation", \(\displaystyle ar^2+ br+ c= 0\), solve that to get "characteristic roots" \(\displaystyle \alpha_1\) and \(\displaystyle \alpha_2\) and then form the the general solution of the differential equation in the form \(\displaystyle C_1e^{\alpha_1 x}+ C_2e^{\alpha_2 x}\) (and variations on that such as sine and cosine, since \(\displaystyle e^{ix}= cos(x)+ i sin(x)\)). If you have never seen something like that, you need to discuss this with your teacher since he/she clearly thinks you have.

One of your equations, X''(x) = λ*X(x), is exactly of that form, X''(x)- λX(x)= 0 with characteristic equation \(\displaystyle r^2- \lambda= 0\). What are the roots of that? What is the general solution and what does X(0)= 0, X(4)= 0 tell you?