The general solution of the differential equation \(\displaystyle\frac{dy}{dx} = \frac{2y}{x + 3}\), \(x > -3\), is (C is an arbitrary constant)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(y = C(x + 3)^2\)
The variables separate:
\( \displaystyle \frac{dy}{y} = \frac{2\,dx}{x + 3} \)
Integrating both sides:
\(\log|y| = 2\log(x + 3) + c\)
\(\log|y| = \log(x + 3)^2 + c\)
Taking the exponential of both sides and writing \(\pm e^{c}\) as C:
\(y = C(x + 3)^2\)
The constant multiplies \((x + 3)^2\); it is not added to it.