Question

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y² = 4ax

Answer 1

The equation of family of curves is \[y^2 = 4ax.........(1)\]
where `a` is a parameter.
As this equation has only one arbitrary constant, we shall get a differential equation of first order.
Differentiating (1) with respect to x, we get
\[2y\frac{dy}{dx} = 4a\]
\[ \Rightarrow 2y\frac{dy}{dx} = \frac{y^2}{x} ...........\left[\text{Using }\left( 1 \right) \right]\]
\[ \Rightarrow 2x\frac{dy}{dx} = y\]
\[ \Rightarrow y - 2x\frac{dy}{dx} = 0\]
It is the required differential equation.