Question

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = ax³

Answer 1

The equation of family of curves is \[y = a x^3.........(1)\]

where `a` is a parameter.

As this equation has only one arbitrary constant, so we shall get a differential equation of first order.

Differentiating (1) with respect to x, we get

\[\frac{dy}{dx} = 3a x^2 \]

\[ \Rightarrow \frac{dy}{dx} = 3 \times \frac{y}{x^3} \times x^2 .........\left[\text{Using}\left( 1 \right) \right]\]

\[ \Rightarrow x\frac{dy}{dx} = 3y\]

It is the required differential equation.