I forgot how to do my ODES Stuck on a PDE question.

[calvino]

Let's say I assumed that the answer to a PDE was U(x,t)= XT, where X,T are functions. I then further my answer by getting to a point for
T'/T=kX''/X, where k is some constant given in the boundary conditions. I then continue by working on either side to find each function. Suppose I work on the right hand side (RHS).

Since i know both LHS and RHS are indepent of each other, I can continue and say that RHS= -$. Now, I check the first case there $<0. So i say that $=-h^2 and get kX'' + (h^2)X=0.

I am stuck here. Does X then turn out to be a function of cosM and sinM, when $<0? (M being theta, or some angle). Furthermore, X=CcosM + DsinM...right?


note: I didnt include that boundary conditions as I doubt that they are needed for my question.

 

[LeonhardEuler]

Does X then turn out to be a function of cosM and sinM, when $<0? (M being theta, or some angle). Furthermore, X=CcosM + DsinM...right?
note: I didnt include that boundary conditions as I doubt that they are needed for my question.

Yes, one form of the solution would be a sum of a sin and cos function. I think you would be better off writing the solution as Acos(mx) + Bsin(mx) rather than cos(M) + sin(M) so that you'll be able to take the derivatives when you plug back into the ODE to find the value of m. Also, I wouldn't bother solving the cases of $>0 and $<0 separately. Just assume the form to be exponential and Euler's formula can put your answer in a more pleasent form.