Problem Statement
Find the general solution of the differential equation:
$$(1 + x^2) \frac{dy}{dx} + 2xy = \frac{1}{1 + x^2}$$
Verified Solution & Marking Scheme
Divide by (1 + x²) to Put in Standard Linear Form
$$\frac{dy}{dx} + \left(\frac{2x}{1 + x^2}\right) y = \frac{1}{(1 + x^2)^2}$$
Here $P(x) = \frac{2x}{1 + x^2}$ and $Q(x) = \frac{1}{(1 + x^2)^2}$.
Calculate Integrating Factor (I.F.)
$$\text{I.F.} = e^{\int \frac{2x}{1 + x^2} dx} = e^{\ln(1 + x^2)} = 1 + x^2$$
Solve by Integration
$$y \cdot (\text{I.F.}) = \int Q(x) \cdot (\text{I.F.}) \, dx + C$$
$$y(1 + x^2) = \int \frac{1}{(1 + x^2)^2} \cdot (1 + x^2) \, dx + C = \int \frac{1}{1 + x^2} \, dx + C$$
$$y(1 + x^2) = \arctan x + C$$
$$y = \frac{\arctan x + C}{1 + x^2}$$