Hi everyone!
Can you please help me solve this ordinary differential equation in the attachment below?
Thank you.
This is a separable ODE.
$\dfrac{y^2-b^2}{y^2+b^2}dy = -\dfrac{x^2+a^2}{x^2-a^2}dx$
$y-b\arctan\left(\dfrac{y}{b}\right) = -x-2\int\left(\dfrac{a^2}{x^2-a^2}\right)dx = -x-2a\text{tanh}^{-1}\left(\dfrac{x}{a}\right)+C$
The final integral can be split up using Partial Fractions if you do not like inverse hyperbolic functions.