Solving Problem with Derivatives as Initial Conditions

[CINA]

Homework Statement

I've been given equations that have derivatives as initial conditions, rather than things like u(0,t)=u(L,t)=0

Things like this:

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Homework Equations

The Attempt at a Solution

I can solve problems with condition like u(0,t)=u(L,t)=0 but how do you solve them with derivatives?

Wouldn't for lamba=0 X''=0 -> X=c1*x+c2 and applying x'(0)=0 make X=c2?

 

[Coto]

You would follow the exact same process (of separation of variables).

The reason $\lambda = 0$ doesn't work is the same reason it doesn't work when you use the standard boundary conditions you've listed above.

The end result is you end up with an equation for X as $X(x) = c_1 \cos{\sqrt{\lambda}x}$. What does this say about $\lambda$? To finish the problem, you need to apply the idea of Fourier series and use your initial condition.