Let $y = y(x)$ be the solution of the differential equation $\left(xy-5x^2\sqrt{1+x^2}\right)dx+\left(1+x^2\right)dy = 0$, $y(0) = 0$. Then $y(\sqrt{3})$ is equal to
Step-by-Step Solution
Key Concept: Rearrange as $\dfrac{dy}{dx}+\dfrac{x}{1+x^2}y = \dfrac{5x^2}{\sqrt{1+x^2}}$; I.F. $= e^{\frac{1}{2}\ln(1+x^2)} = \sqrt{1+x^2}$; integrate the right side directly.
$\dfrac{dy}{dx}+\dfrac{xy}{1+x^2} = \dfrac{5x^2}{\sqrt{1+x^2}}$. I.F. $= \sqrt{1+x^2}$.
$y\sqrt{1+x^2} = \int\dfrac{5x^2\cdot\sqrt{1+x^2}}{\sqrt{1+x^2}}dx = \int 5x^2\,dx = \dfrac{5x^3}{3}+C$.
At $x=0$, $y=0$: $C=0$. So $y = \dfrac{5x^3}{3\sqrt{1+x^2}}$.
$$y(\sqrt{3}) = \frac{5\cdot 3\sqrt{3}}{3\cdot 2} = \frac{5\sqrt{3}}{2}.$$
Correct Answer: 2