Question

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

Answer 1

The equation of family of curves is \[y = e^{ax} \]
\[ \Rightarrow \log y = ax .........\left( 1 \right)\]

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

\[\frac{1}{y}\frac{dy}{dx} = a\]

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

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

It is the required differential equation.