User:EGM6321.f11.team1.pan/HW7

= R*7.7 Find the separated equation for the Laplace Equation in Parabolic coordinates= == Given == The Laplacian in parabolic coordinates given by
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$$
 * $$\displaystyle \nabla^2 u=\frac{1}{\mu^2+\nu^2}\left(\frac{\partial^2 u}{\partial \mu^2}+\frac{\partial^2 u}{\partial \nu^2}\right)$$
 * $$\displaystyle (Equation\;7.7.1)
 * $$\displaystyle (Equation\;7.7.1)
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Find
Find the separated equation for the Laplace equation in Pararbolic coordinates.

Solution
Separation of variables: $$\displaystyle u(\mu,\nu)=U(\mu)V(\nu)$$ Substituting it into Equation (7.7.1), $$\displaystyle \Delta u=0=\frac{1}{\mu^2+\nu^2}\left(V \frac{\partial^2 U}{\partial \mu^2}+ U \frac{\partial^2 V}{\partial \nu^2}\right)$$ Cancel $$\displaystyle \frac{1}{\mu^2+\nu^2} $$ and $$\displaystyle UV$$, $$\displaystyle \frac{1}{U}\,\frac{\partial^2 U}{\partial \mu^2}+ \frac{1}{V}\,\frac{\partial^2 V}{\partial \nu^2}=0$$ Rearranging it, we can obtain that $$\displaystyle \frac{1}{U}\,\frac{\partial^2 U}{\partial \mu^2} = - \frac{1}{V}\,\frac{\partial^2 V}{\partial \nu^2}=k$$ Therefore, the separated equations are: