x^2 \frac{dy}{dx} + y^2 = xy

Steps to Solve

1. Rearrange the Differential Equation

   We start with the given equation:
   $$ x^2 \frac{dy}{dx} + y^2 = xy $$
   Rearranging it yields:
   $$ x^2 \frac{dy}{dx} = xy - y^2 $$

2. Factor out (y)

   On the right-hand side, we can factor (y):
   $$ x^2 \frac{dy}{dx} = y(x - y) $$

3. Separate Variables

   Now, we'll separate the variables (y) and (x):
   $$ \frac{dy}{y(x - y)} = \frac{dx}{x^2} $$

4. Integrate Both Sides

   Next, integrate both sides:

   • Left-hand side:
     $$ \int \frac{dy}{y(x - y)} $$
   • Right-hand side:
     $$ \int \frac{dx}{x^2} $$

   We can rewrite the right side as:
   $$ \int x^{-2} dx = -\frac{1}{x} + C_1 $$

5. Solve the Left Side Integral (Using Partial Fractions)

   The left-hand side integral can be solved using partial fractions:
   $$ \frac{1}{y(x - y)} = \frac{A}{y} + \frac{B}{x - y} $$

   After integrating, we obtain:
   $$ A \ln |y| - B \ln |x - y| = -\frac{1}{x} + C_1 $$

6. Combine and Rearrange

   Finally, combine and solve for (y) in terms of (x). This will involve exponentiation and may require simplification based on the constants.