Let y=y(x) be the solution of (xy-5x^2√1+x^2),dx+(1+x^2),dy=0, y(0)=0.

Question

Let $y=y(x)$ be the solution of $(xy-5x^2\sqrt{1+x^2})\,dx+(1+x^2)\,dy=0$, $y(0)=0$. Then $y(\sqrt3)$ is:

Accepted Answer

$$\frac{dy}{dx}+\frac{xy}{1+x^2} = \frac{5x^2}{\sqrt{1+x^2}}.$$

I.F. $=\sqrt{1+x^2}$.
$$y\sqrt{1+x^2} = \int 5x^2\,dx = \frac{5x^3}{3}+C.$$

Using $y(0)=0 \Rightarrow C=0$. So $y=\dfrac{5x^3}{3\sqrt{1+x^2}}$. At $x=\sqrt3$: $y=\dfrac{5\cdot3\sqrt3}{3\cdot2}=\dfrac{5\sqrt3}{2}$.

Solutions · 1