The general solution of the differential equation \(y\displaystyle\frac{dy}{dx} = e^{2x}\) is (C is an arbitrary constant)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(y^2 = e^{2x} + C\)
The variables are already separated:
\( y\,dy = e^{2x}\,dx \)
Integrating both sides:
\( \displaystyle \frac{y^2}{2} = \frac{e^{2x}}{2} + C_1 \)
Multiplying by 2 and writing C for \(2C_1\):
\( y^2 = e^{2x} + C \)
Why the other options are incorrect
Option (1) \(y^2 = 2e^{2x} + C\) — Incorrect. Differentiating gives \(2y\frac{dy}{dx} = 4e^{2x}\), that is \(y\frac{dy}{dx} = 2e^{2x}\).
Option (2) \(y = e^{x} + C\) — Incorrect. Here \(y\frac{dy}{dx} = (e^{x} + C)e^{x}\), which equals \(e^{2x}\) only when C = 0, so this family does not solve the equation for every C.
Option (3) \(y^2 = \displaystyle\frac{e^{2x}}{2} + C\) — Incorrect. Only the left side was doubled. Differentiating gives \(y\frac{dy}{dx} = \frac{e^{2x}}{2}\).