Question

Solve the differential equation: \[ (1 + x^2) \frac{dy}{dx} + 2xy = 4x^2. \]

Hint:

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

Answer 1

Step 1: Rewrite the equation in standard form. \[ \frac{dy}{dx} + \frac{2xy}{1 + x^2} = \frac{4x^2}{1 + x^2}. \] Step 2: Identify integrating factor (IF). \[ \text{IF} = e^{\int \frac{2x}{1 + x^2} dx}. \] Let \( u = 1 + x^2 \), so \( du = 2x dx \): \[ \int \frac{2x}{1 + x^2} dx = \log |1 + x^2|. \] \[ \text{IF} = e^{\log |1 + x^2|} = 1 + x^2. \] Step 3: Multiply both sides by the integrating factor. \[ (1 + x^2) \frac{dy}{dx} + 2xy = 4x^2. \] \[ \frac{d}{dx} [y(1 + x^2)] = 4x^2. \] Step 4: Integrate both sides. \[ \int d(y(1 + x^2)) = \int 4x^2 dx. \] \[ y(1 + x^2) = \frac{4}{3} x^3 + C. \] Step 5: Solve for \( y \). \[ y = \frac{4}{3} \frac{x^3}{1 + x^2} + \frac{C}{1 + x^2}. \] Final Solution: \[ y = \frac{4x^3}{3(1 + x^2)} + \frac{C}{1 + x^2}. \]