Question

The differential equation whose solution is \(y = e^{ax}\) is

• A. \( y \dfrac{dy}{dx} = x \log y \)
• B. \( \dfrac{dy}{dx} = x \log x \)
• C. \( \dfrac{dy}{dx} = y \log x \)
• D. \( x \dfrac{dy}{dx} = y \log y \)

Hint:

To form a differential equation, eliminate arbitrary constants from the given solution.

Answer 1

The Correct Option is D

Step 1: Take logarithm of the given solution.
Given \(y = e^{ax}\), taking natural logarithm on both sides: \[ \log y = ax \]
Step 2: Differentiate with respect to \(x\).
\[ \frac{1}{y}\frac{dy}{dx} = a \]
Step 3: Eliminate the constant \(a\).
From \(\log y = ax\), we get \[ a = \frac{\log y}{x} \]
Step 4: Substitute the value of \(a\).
\[ \frac{1}{y}\frac{dy}{dx} = \frac{\log y}{x} \Rightarrow x\frac{dy}{dx} = y\log y \]