For the differential equation \(\displaystyle\frac{dy}{dx} = -3y\), the function \(y = 4e^{-3x}\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → a particular solution
First check that it is a solution. Differentiating \(y = 4e^{-3x}\):
\( \displaystyle \frac{dy}{dx} = -12e^{-3x} = -3\left(4e^{-3x}\right) = -3y \)
This holds for every real x, so it is a solution for all x, not only for x > 0.
A general solution contains as many arbitrary constants as the order (here one, as in \(y = Ce^{-3x}\)). A solution with no arbitrary constant is a particular solution.
\(y = 4e^{-3x}\) has no arbitrary constant (C has been given the value 4), so it is a particular solution.