Question

𝑦 + 𝑑 𝑑𝑥 ( 𝑥𝑦 ) = 𝑥 (sin 𝑥 + 𝑥)

Answer 1

$y = -\cos x + \frac{2\sin x}{x} + \frac{2\cos x}{x^2} + \frac{x^2}{4} + \frac{C}{x^2}$

Answer 2

The given differential equation is transformed into the standard form of a first-order linear ODE: $\frac{dy}{dx} + \frac{2}{x}y = \sin x + x$. The integrating factor is calculated as $x^2$. Multiplying the equation by $x^2$ results in $\frac{d}{dx}(yx^2) = x^2\sin x + x^3$. Integrating both sides yields $yx^2 = \int x^2\sin x dx + \int x^3 dx$. Evaluating these integrals gives $yx^2 = -x^2\cos x + 2x\sin x + 2\cos x + \frac{x^4}{4} + C$. Solving for $y$ provides the general solution.