The solution of the differential equation \(\displaystyle\frac{dy}{dx} = 2y\) satisfying \(y(0) = 5\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(y = 5e^{2x}\)
The equation is variables separable:
\( \displaystyle \frac{dy}{y} = 2\,dx \)
Integrating both sides:
\(\log|y| = 2x + C_1\)
\(y = Ce^{2x}\)
Putting x = 0, y = 5:
\(5 = Ce^{0} = C\)
\(y = 5e^{2x}\)
Why the other options are incorrect
Option (1) \(y = e^{2x} + 4\) — Incorrect. It gives y(0) = 5, but \(\frac{dy}{dx} = 2e^{2x}\) is not \(2y = 2e^{2x} + 8\).
Option (3) \(y = 5e^{x/2}\) — Incorrect. Here \(\frac{dy}{dx} = \frac{y}{2}\), not 2y.
Option (4) \(y = 5e^{-2x}\) — Incorrect. Here \(\frac{dy}{dx} = -2y\), the wrong sign.