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how to deal with this type of elliptic equation? $$v_{\xi\xi}+v_{\eta\eta}-4\tan{\xi}.v_{\xi}=0$$ Can we apply separation of variables method here? I have tried but it seems complicated?

Godgog Arsenal
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$$u_{xx}+u_{yy}-4\tan(x)u_x=0$$ $$u(x,y)=X(x)Y(y)\quad\to\quad X''Y+XY''-4\tan(x)X'Y=0$$

$$\frac{X''}{X}+\frac{Y''}{Y}-4\tan(x)\frac{X'}{X}=0$$

$$\frac{Y''}{Y}=-\frac{X''}{X}+4\tan(x)\frac{X'}{X}=\lambda=\text{constant} $$ $$\begin{cases}Y''(y)-\lambda Y(y)=0 \\X''(x)-4\tan(x)X'(x)+\lambda X(x)=0\end{cases}$$ The first ODE is easy. The second involves hypergeometric functions.

JJacquelin
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  • Thanks alot, that's what I obtained, however, I am struggling with the second ODE! I have never faced this type of ODE – Godgog Arsenal May 20 '17 at 09:15
  • If you don't know the hypergeometric functions (in the present case : the Gauss hypergeometric function $_2F_1$) you cannot express the solutions of the ODE on a closed form. You can solve the ODE on the form of infinite series, namely the hypergeometric series. In practice, one can use numerical calculus. – JJacquelin May 20 '17 at 09:40
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    Nevertheless, in some particular cases, the hypergeometric functions can be simplified to functions of lower level, sometimes to elementary functions. But this is not possible in the general case. There is a short table pp.26-27 in the paper : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales – JJacquelin May 20 '17 at 09:40
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    For example, in the particular cases $\lambda=4(n^2-1)$ and $n$ integer, you can obtain a lot of particular solutions with elementary functions. – JJacquelin May 22 '17 at 06:55