The differential equation satisfied by the equation \( y=ae^{bx} \), where \( a \) and \( b \) are arbitrary constants, is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( y\ \displaystyle\frac{d^2y}{dx^2}=\left(\displaystyle\frac{dy}{dx}\right)^2 \)
There are two arbitrary constants, so the equation comes from differentiating twice and eliminating a and b:
\( \displaystyle \frac{dy}{dx}=abe^{bx}=by \)
\( \displaystyle \frac{d^2y}{dx^2}=b\frac{dy}{dx} \)
From the first, \( b=\frac{1}{y}\frac{dy}{dx} \). Substituting in the second:
\( \displaystyle \frac{d^2y}{dx^2}=\frac{1}{y}\left(\frac{dy}{dx}\right)^2 \)
\( \displaystyle y\frac{d^2y}{dx^2}=\left(\frac{dy}{dx}\right)^2 \)