solve the following differential equation?

Question

solve the following differential equation?

x² y' = y² + x y - x²

and y (1) = 2

Please answer with steps

Answer 1

If you divide the equation by x², you have dy/dx = (y/x)² + (y/x) - 1

Then if you let u = y/x, you change the equation to du/(u² - 1) = dx/x

After solving this equation, you have (c = integration constant) u = (1 + c²x²)/(1 - c²x²) = y/x

Then, solving for the initial condition, y = x(1 + x²/3)/(1 - x²/3)