\(\frac{dy}{dx} = \frac{5x^2\sqrt{1 + x^2} - xy}{1 + x^2}
\)
\(\frac{dy}{dx} + \frac{x}{1 + x^2} y = \frac{5x^2\sqrt{1 + x^2}}{1 + x^2}\)
\(\frac{dy}{dx} + \frac{x}{1 + x^2} y = 5x^2 \frac{1}{\sqrt{1 + x^2}}\)
first-order linear differential equation
\(\frac{dy}{dx} + P(x)y = Q(x)\)
IF = \(e^{\int P(x) dx} = e^{\int \frac{x}{1 + x^2} dx}\)
\(u = 1 + x^2 \)
\(du = 2x dx\)
IF = \(e^{\frac{1}{2} \ln |1 + x^2|} = (1 + x^2)^{1/2}\)
\(y \cdot \text{IF} = \int (\text{IF} \cdot Q) \, dx + C\)
\(y(1 + x^2)^{1/2} = \int 5x^2 dx = \frac{5}{3} x^3 + C\)
y(0) = 0 , C=0
\(y = \frac{5x^3}{3\sqrt{1 + x^2}}\)
\(y(\sqrt{3}) = \frac{5\sqrt{3}}{2}\)
