Question

Form a differential equation to the family of curves $y^2=4a(x+a)$, where $a$ is an arbitrary constant.

Answer 1

Given $y^2=4a(x+a)$

differentiating,

$=2y\frac{dy}{dx}=4a$

$\Rightarrow y\frac{dy}{dx}=2a$

differentiating wrt x

Let y = u $\Rightarrow \frac{du}{dx}=\frac{dy}{dx}$

$v=\frac{dy}{dx}$ $\frac{dv}{dx}=\frac{d^{2}y}{d{x}^2}$

By product formula,

$y\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}.\frac{dy}{dx}=0$

$\Rightarrow y$ $y^{11}+(y^1{)}^2=0$