The differential equation whose solution is $y=e^{a x}$ is
The differential equation whose solution is $y=e^{a x}$ is
$y \frac{d y}{d x}=x \log y$
$\frac{d y}{d x}=x \log x$
$\frac{d y}{d x}=y \log x$
$x \frac{d y}{d x}=y \log y$
Solution
Given $y=e^{a x}$
Taking log on both sides, we get
$\therefore \frac{\log y}{y}=\log e^{a x} \Rightarrow \log y=\operatorname{axlog} e \Rightarrow \log y=a x$ ...(1)
$\therefore \frac{1}{y} \frac{d y}{d x}=a$
Substituting value of ' $a$ ' in equation (1), we get
$\log y=\left[\left(\frac{1}{y}\right) \frac{d y}{d x}\right] x \Rightarrow x \frac{d y}{d x}=y \log y$