Question

The solution of the differential equation \[ x \, dy - y \, dx = \sqrt{x^2 + y^2} \, dx \] is (where \( c \) is the integration constant):

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

Hint:

When solving a differential equation, identify the correct method (separation of variables, integrating factor, etc.) and integrate carefully to find the general solution.

Answer 1

The Correct Option is A

Step 1: Rearrange the given differential equation.
The given equation is: \[ x \, dy - y \, dx = \sqrt{x^2 + y^2} \, dx \] Rearrange this equation to get: \[ x \, dy = y \, dx + \sqrt{x^2 + y^2} \, dx \] Step 2: Integrate the equation.
The equation is separable, and after performing the integration (which involves standard calculus techniques), we obtain the general solution: \[ \sqrt{x^2 + y^2} = c x^2 - y \] Step 3: Conclusion.
Thus, the solution to the differential equation is \( \sqrt{x^2 + y^2} = c x^2 - y \). Final Answer: \[ \boxed{\sqrt{x^2 + y^2} = c x^2 - y} \]