If \(y = y(x)\), \(y \gt 0\), is the solution of the differential equation \(\displaystyle\frac{dy}{dx} = 2\sqrt{y}\) satisfying \(y(0) = 1\), then \(y(2)\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 9
Separating the variables:
\( \displaystyle \frac{dy}{\sqrt{y}} = 2\,dx \)
Integrating both sides:
\( 2\sqrt{y} = 2x + C_1 \)
\( \sqrt{y} = x + C \)
Using \(y(0) = 1\):
\( 1 = 0 + C \), so \( C = 1 \)
The particular solution is
\( y = (x + 1)^2 \)
At x = 2:
\( y(2) = 3^2 = 9 \)