The solution of the differential equation \(\displaystyle\frac{dy}{dx} = \sqrt{\frac{y}{x}}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\sqrt{x} - \sqrt{y} = C\): C is an arbitrary constant
The equation is in variables separable form, since \(\sqrt{\frac{y}{x}} = \frac{\sqrt{y}}{\sqrt{x}}\):
\( \displaystyle \frac{dy}{\sqrt{y}} = \frac{dx}{\sqrt{x}} \)
Integrating both sides:
\( \displaystyle \int y^{-1/2}\,dy = \int x^{-1/2}\,dx \)
\(2\sqrt{y} = 2\sqrt{x} + C_1\)
Dividing by 2 and rearranging, with \(C = -\frac{C_1}{2}\):
\(\sqrt{x} - \sqrt{y} = C\)