Question

The solution of the differential equation \[ y \frac{dy}{dx} = x \left[ \frac{y^2}{x^2} + \varphi \left( \frac{y^2}{x^2} \right) \right] \] is:

• A. \( \varphi \left( \frac{y^2}{x^2} \right) = Cx \)
• B. \( x \frac{y^2}{x^2} = C \)
• C. \( \left( \frac{y^2}{x^2} \right) = Cx^2 \)
• D. \( x \varphi \left( \frac{y^2}{x^2} \right) = C \)

Hint:

To solve differential equations, first try to simplify them and then use techniques like separation of variables or substitution to solve them.

Answer 1

The Correct Option is C

We are given the differential equation: \[ y \frac{dy}{dx} = x \left[ \frac{y^2}{x^2} + \varphi \left( \frac{y^2}{x^2} \right) \right] \] Step 1: Rewrite the equation First, simplify the equation by dividing both sides by \( x \): \[ y \frac{dy}{dx} = \frac{y^2}{x} + \varphi \left( \frac{y^2}{x^2} \right) \] Step 2: Apply separation of variables Now, rearrange the equation to separate variables: \[ \frac{dy}{dx} = \frac{y}{x} + \frac{1}{x} \varphi \left( \frac{y^2}{x^2} \right) \] Step 3: Solve the equation By solving the equation, we get: \[ \left( \frac{y^2}{x^2} \right) = Cx^2 \] Thus, the correct answer is \( \left( \frac{y^2}{x^2} \right) = Cx^2 \).