Question

Form the differential equation corresponding to y = e^mx by eliminating m.

Answer 1

The equation of the family of curves is \[y = e^{mx}.................(1)\]

where \[m\] is a parameter.

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

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

\[\frac{dy}{dx} = m e^{mx} \]

\[ \Rightarrow \frac{dy}{dx} =my.............\left[\text{ Using equation}(1)\right]\]

\[\Rightarrow m = \frac{1}{y}\frac{dy}{dx}.................(2)\]

Now, from equation (1), we get

\[\ln y = \ln e^{mx} \]
\[ \Rightarrow \ln y = mx \ln e\]
\[ \Rightarrow \ln y = mx\]
\[ \Rightarrow m = \frac{1}{x}\ln y .................(3)\]
Comparing equations (2) and (3), we get
\[\frac{1}{x}\ln y = \frac{1}{y}\frac{dy}{dx}\]
\[ \Rightarrow x\frac{dy}{dx} = y \ln y\]

It is the required differential equation.