(adsbygoogle = window.adsbygoogle || []).push({}); 1. verify that u(t,x,y)=e^{-λt}sin(αt)cos(βt) (for arbitrary α, β and with λ=α^{2}+β^{2}) satisfies the 2-D Heat Equation.

2. u_{t}=Δu

3. I began with:

Δu=u_{xx}+u_{yy}.

note the equation does not contain variable "x"

so u_{xx}=0 i.e. Δu=u_{yy}

u_{y}=e^{-λt}sin(αt){-βsin(βt)}

u_{yy}=e^{-λt}sin(αt){-β^{2}cos(βt)}

next I found u_{t}

u_{t}=cos(βy) {e^{-λt}αcos(αt)+sin(αt)-λe^{-λt}}

I have tried to reduce both equations but don't see how they are equal. I also have tried using the λ=α^{2}+β^{2}to re-write the eq. Any suggestions?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Verification of solution to Heat Equation

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