Question

The particular solution of the differential equation \(xdy+2ydx = 0\), when \(x = 2\) and \(y = 1\) is

• A. \(x\sqrt{y} = 2\)
• B. \(x\sqrt{y} = -2\)
• C. \(\sqrt{x}y = 2\)
• D. \(x\sqrt{y} = 1\)

Hint:

Separate the variables and integrate.

Answer 1

The Correct Option is A

Step 1: Separate
\(x\,dy=-2y\,dx\) gives \(\frac{dy}{y}=-2\frac{dx}{x}\).

Step 2: Integrate
\(\log y=-2\log x+\log C\), so \(yx^2=C\).

Step 3: Use the condition
At \(x=2\), \(y=1\): \(C=4\). So \(x^2y=4\), and taking positive roots, \(x\sqrt y=2\). Option (A).

Final Answer:
The solution is \(x\sqrt y=2\), option (A). \[ \boxed{\text{(A)}} \]