Question

Form the differential equation from the following primitive where constants are arbitrary:
y² = 4ax

Answer 1

The equation of family of curves is \[y^2 = 4ax.................(1)\]

where a is an arbitrary constant.

This equation contains only one arbitrary constant, so we shall get a differential equation of first order.

Differentiating equation (1) with respect to x, we get

\[2y\frac{dy}{dx} = 4a\]

\[ \Rightarrow \frac{y}{2}\frac{dy}{dx} = a .................\left( 2 \right)\]

Putting the value of  a  in equation (1), we get

\[y^2 = 4\frac{y}{2}\frac{dy}{dx}x\]

\[ \Rightarrow y = 2x\frac{dy}{dx}, \]

It is the required differential equation.