# Thread: First order Differential Equations

1. ## First order Differential Equations

Hi,

Can someone please tell me how to solve for the following question:

Calculate ans sketch the orthogonal trajectories for the hyperbolas: x^2 - y^2 = k

I am quite lost in this topic and so not sure where to start off, may someone please show me the step on how to do the problem?
Thank you.

2. Originally Posted by zxcv
Hi,

Can someone please tell me how to solve for the following question:

Calculate ans sketch the orthogonal trajectories for the hyperbolas: x^2 - y^2 = k

I am quite lost in this topic and so not sure where to start off, may someone please show me the step on how to do the problem?
Thank you.
Using implicit differentiation: $\displaystyle 2x - 2y \frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = \frac{x}{y}$.

Therefore the differential equation $\displaystyle \frac{dy}{dx} = - \frac{y}{x}$ will have as its solution the family of orthogonal trajectories to $\displaystyle x^2 - y^2 = k$.