Hello

Problem:

Find the orthogonal trajectories of the following family of curves:

$\displaystyle x^n+y^n=a^n$ ; n held fixes, $\displaystyle n \neq 2$

Solution:

Differentiate with respect to x:

$\displaystyle nx^{n-1}+ny^{n-1} \cdot \dfrac{dy}{dx}=0$

devide by n and re-arrange:

$\displaystyle \dfrac{dy}{dx} = - \dfrac{x^{n-1}}{y^{n-1}}$

the equation for orthogonal family:

$\displaystyle \dfrac{dy}{dx} = \dfrac{y^{n-1}}{x^{n-1}}$

which is separable:

$\displaystyle y^{1-n} dy = x^{1-n} dx$

Solving:

$\displaystyle y^{2-n} = x^{2-n} + c$ .. which is the equation of orthogonal trajectories ..

any mistakes?