Question

See next comment, more diffeq's!

Answer 1

This was a question on my test, and did not resemble anything we've done before.

Solve the differential equation using fourier transforms given:\[u_{xx}+u_{yy}+u=0\]\[\lim_{y \rightarrow \infty}\hat{u}(\xi,y)=0\]

After transforming:
\[-\xi ^2 \hat{u}(\xi,y)+\hat{u}_{yy}(\xi,y)+\hat{u}(\xi,y)=0\]Let\[\hat{u}(\xi,y)=r(y)\]So,\[-\xi^2r(y)+r''(y)+r(y)=0\]Rearranging:\[r''(y)+(-\xi^2+1)r(y)=0\]

I imagine now you would test three cases for when\[(-\xi^2+1)\]Is positive, negative, and equal to zero.... But I've never seen something like:\[\lim_{y \rightarrow \infty}\hat{u}(\xi,y)=0\]as a condition before in class or on hw.

Answer 2

@mahmit2012

Answer 3

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