Question

Solve the differential equation: \[ x^2 \frac{dy}{dx} = 2xy. \]

Hint:

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

Answer 1

Rewrite the equation: \[ x^2 \frac{dy}{dx} = 2xy \implies \frac{dy}{dx} = \frac{2y}{x}. \] This is a separable differential equation. Separate variables: \[ \frac{dy}{y} = \frac{2}{x} dx. \] Integrate both sides: \[ \int \frac{1}{y} dy = \int \frac{2}{x} dx \implies \ln |y| = 2 \ln |x| + C, \] where \( C \) is the constant of integration. Rewrite: \[ \ln |y| = \ln |x|^2 + C \implies y = e^{C} x^{2} = K x^{2}, \quad K = e^{C}. \] % Final solution: \[ \boxed{y = K x^{2}}. \]