Question

The solution of the differential equation \[ \left(1 + x \sqrt{x^2 + y^2}\right) dx + \left( \sqrt{x^2 + y^2} - 1 \right) y dy = 0 \]

• A. \( x^2 + y^2 + \frac{1}{3} \left( x^2 + y^2 \right)^{3/2} = C \)
• B. \( x^2 + y^2 - \frac{1}{2} \left( x^2 + y^2 \right)^{1/2} = C \)
• C. \( x^2 + y^2 + \frac{1}{3} \left( x^2 + y^2 \right)^{3/2} = C \)
• D. None of these

Hint:

When solving a differential equation, try to identify a method of simplification such as substitution to reduce it to a more manageable form.

Answer 1

The Correct Option is A

Solving the differential equation step by step gives us the required solution as \( x^2 + y^2 + \frac{1}{3} \left( x^2 + y^2 \right)^{3/2} = C \).
Final Answer: \[ \boxed{x^2 + y^2 + \frac{1}{3} \left( x^2 + y^2 \right)^{3/2} = C} \]