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
$$\begin{gathered} \frac{dy}{dx}=m{e}^{mx} \\ \Rightarrow \frac{dy}{dx}=my[\mathrm{Using}\mathrm{equation}(1\left)\right] \end{gathered}$$
$\Rightarrow m=\frac{1}{y}\frac{dy}{dx}$ ...(2)
Now, from equation (1), we get
$$\begin{gathered} \ln y=\ln e^{mx} \\ \Rightarrow \ln y=mx\ln e \\ \Rightarrow \ln y=mx \\ \Rightarrow m=\frac{1}{x}\ln y...\left(3\right) \end{gathered}$$
Comparing equations (2) and (3), we get
$$\begin{gathered} \frac{1}{x}\ln y=\frac{1}{y}\frac{dy}{dx} \\ \Rightarrow x\frac{dy}{dx}=y\ln y \end{gathered}$$
It is the required differential equation.