A PDE I can't solve by seperation of variables

NapoleonZ
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Homework Statement





Homework Equations


After simplification, the PDE is

(b^2/a^2)(d^2 v/ d x^2) + (d^2 v/ d y^2) = -1

The Attempt at a Solution


Obviously, it can't be solved by separation of variables. And I also failed in similarity solution.
 
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Are a and b constants? If so, you need to change the x-scale: x = x' * b/a and get the standard Poisson's equation.
 
quZz said:
Are a and b constants? If so, you need to change the x-scale: x = x' * b/a and get the standard Poisson's equation.

Yes, they are.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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