# Thread: Orthogonal Trajectories .. Check my solution

1. ## Orthogonal Trajectories .. Check my solution #1

Hello

Problem:
Find the orthogonal trajectories of the following family of curves:
$x^n+y^n=a^n$ ; n held fixes, $n \neq 2$

Solution:
Differentiate with respect to x:

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

devide by n and re-arrange:

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

the equation for orthogonal family:

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

which is separable:

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

Solving:

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

any mistakes?

2. looks good to me.

3. Thanks.

4. you already do it by clicking on the "thanks button," there's no need you to post a "thanks" again, it's enough with clicking the button.

5. thanks again .
I'll do it.