Thread: Derivatives of double integrals help please! :(

Hi,
Just wondering if anyone can help, I've been given a question:

Verify that the solution to the inhomogeneous problem
w_t-w_xx=f(x,t) t>0
w(x,0)=0
is
w(x,t)=int _0^t int_-infty^infty f(s,v)z(x-s,t-v) dsdv

I know to verify this I need to differentiate the double integral but I'm just stuck on how to go about it as opposed to a single inetgral.

Thanks!

I am not familiar with the formatting that you have typed so I apologize that I cannot give you a more precise answer. Although to undo a double integral "verify", you will need to Partial Differentiate. With the provided equation w(x,t)=int _0^t int_-infty^infty f(s,v)z(x-s,t-v)dsdv, we can see that you will be integrating with respect to "s" first while holding "v" as a constant. You will do the same when you differentiate; first differentiate the function as you normally would for a single variable, with respect to "s", and hold "v" as a constant. Then you do the same but with respect to "v" while holding "s" as a constant. Hope that helps...