Question

Find the general solution of the differential equation \( x \frac{dy}{dx} + 2y = x^2 \); \( x \neq 0 \).

Hint:

For linear first-order differential equations, use the integrating factor method: \( IF = e^{\int P(x)dx} \).

Answer 1

Rewriting: \[ \frac{dy}{dx} + \frac{2y}{x} = x. \] This is a linear first-order differential equation. Using the integrating factor \( IF = e^{\int \frac{2}{x} dx} = x^2 \), the solution is: \[ y \cdot x^2 = \int x^3 dx = \frac{x^4}{4} + C. \] Thus, the general solution is: \[ y = \frac{x^2}{4} + \frac{C}{x^2}. \]