The eigenvalues must satisfy:

The eigenfunction is:

Is this much correct?

Thanks.

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- Mar 16th 2011, 06:47 PMdwsmithIBV

The eigenvalues must satisfy:

The eigenfunction is:

Is this much correct?

Thanks. - Mar 19th 2011, 03:54 PMdwsmith

I have shown .

Before I solve for is u(x,t) correct?

Thanks. - Mar 19th 2011, 04:31 PMHallsofIvy
Where did and suddenly come from?

Are they and ?

If so, you cannot, generally, do that. But here it does not hurt.

Quote:

Then . Again, the exponential is not 0 so we must have either B= 0, which would mean the solution was identically equal to 0, and so not an eigenfunction, or cos(L\sqrt{\lambda})= 0.

Quote:

The eigenvalues must satisfy:

Quote:

The eigenfunction is:

Is this much correct?

Thanks.

- Mar 19th 2011, 04:33 PMdwsmith
My book solves all PDEs in the same manner. I don't know why it is done that way but it works in all cases.

Here is what the book says about 1 and 2.

The theorem in the theory of ODE which explicitly states the proposition above is: if the coefficients are continuous on the interval [a,b] and is different from zero throughout the interval, then for any fixed in the interval [a,b] and for any constants . The DE has one and only one solution satisfying at the initial conditions .

Moreover, and are continuous and continuously differentiable functions of both x for and all lambda.

It is convenient to single out the two special solutions of which satisfy the DE

.

We call 1 and 2 the basic solutions at x nought. - Mar 19th 2011, 08:32 PMdwsmith

- Mar 20th 2011, 12:42 PMdwsmith
For the case

If everything above is correct, can I do this?

- Mar 21st 2011, 04:11 PMdwsmith
Ok, I pretty sure the above answer is wrong.

I now have:

How do I show this is true for all x with the IC: