Question

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x² + y² = ax³

Answer 1

The equation of family of curves is \[x^2 + y^2 = a x^3........(1)\]

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

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

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

\[ \Rightarrow 2x + 2y\frac{dy}{dx} = 3\frac{x^2 + y^2}{x}\]

\[ \Rightarrow 2 x^2 + 2xy\frac{dy}{dx} = 3 x^2 + 3 y^2 \]

\[ \Rightarrow 2xy\frac{dy}{dx} = x^2 + 3 y^2 \]

It is the required differential equation.