18.7.4.1 Solution expclass-a

Question:

Convert the equation

$$11 u_{x'x'} + u_{y'y'} = \frac{3}{\sqrt{10}} u_{x'} - \frac{... ...} u_{y'} + \frac{3}{\sqrt{10}} x' - \frac{1}{\sqrt{10}} y' + 1$$

to be as close as possible to the Laplace equation.

Answer:

Define $\xi$ $\vphantom0\raisebox{1.5pt}{$=$}$ $x'/\sqrt{11}$, $\eta$ $\vphantom0\raisebox{1.5pt}{$=$}$ $y'$ to give

$$u_{\xi\xi} + u_{\eta\eta} = \frac{3}{\sqrt{110}} u_{\xi} - \... ...frac{3\sqrt{11}}{\sqrt{10}} \xi - \frac{1}{\sqrt{10}} \eta + 1$$

Then define $v$ by the relation $u$ $\vphantom0\raisebox{1.5pt}{$=$}$ $v e^{a\bar\xi +b\bar\eta}$. Plug it in and you see that for the values of $a$ and $b$ for which the first order derivatives vanish,

$$v_{\xi\xi} + v_{\eta\eta} = \frac{1}{22} v + \left(\frac{3\s... ...qrt{10}}\eta + 1\right) e^{(\eta\sqrt{11}-3\xi)/(2\sqrt{110})}$$