Question

The solution of the differential equation \[ x^2 \frac{dy}{dx} = y^2 + xy \]

• A. \( \frac{x}{y} + \log |x| = c \)
• B. \( \frac{y}{x} + \log |x| = c \)
• C. \( \frac{x}{y} - \log |x| = c \)
• D. \( \frac{y}{x} - \log |x| = c \)

Hint:

For separable differential equations, always separate the variables and integrate both sides to find the solution.

Answer 1

The Correct Option is A

Step 1: Rewriting the equation.
We are given the equation \( x^2 \frac{dy}{dx} = y^2 + xy \). First, we rearrange it to separate the variables: \[ \frac{dy}{dx} = \frac{y^2 + xy}{x^2} \] We can now express this as: \[ \frac{dy}{dx} = \frac{y(y + x)}{x^2} \]
Step 2: Integrating both sides.
By solving this equation using the method of separation of variables, we get the integral: \[ \int \frac{dy}{y(y + x)} = \int \frac{dx}{x} \] After solving this, we obtain the solution \( \frac{x}{y} + \log |x| = c \). This corresponds to option (A).

Step 3: Conclusion.
Thus, the correct solution is \( \frac{x}{y} + \log |x| = c \), making option (A) the correct answer.