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

Question

Let $y=y(x)$ be the solution of the differential equation $(xy-5{x}^{2}\sqrt{1+{x}^{2}})dx+(1+{x}^{2})dy=0,y(0)=0$. Then $y(\sqrt{3})$ is equal to

Accepted Answer

$\begin{gathered} (1+{x}^{2})\frac{dy}{dx}+xy=5{x}^{1}\sqrt{1+{x}^{2}} \ \frac{dy}{dx}+\frac{xy}{1+{x}^{2}}=\frac{5{x}^{2}}{\sqrt{1+{x}^{2}}} \ 	herefore ={e}^{\int \frac{x}{1+{x}^{2}}dx}={e}^{\frac{\ln (1+{x}^{2})}{2}}=\sqrt{1+{x}^{2}} \ 	herefore y\sqrt{1+{x}^{2}}=\int \frac{5{x}^{2}}{\sqrt{1+{x}^{2}}}\cdot \sqrt{1+{x}^{2}}dx \ 	herefore y\sqrt{1+{x}^{2}}=\int \frac{5{x}^{2}}{\sqrt{1+{x}^{2}}}\cdot \sqrt{1+{x}^{2}}dx \ y\sqrt{1+{x}^{2}}=\frac{5{x}^{3}}{3}+C \ \because y(0)=0\Rightarrow 0=0+C\Rightarrow C=0 \ 	herefore y=\frac{5{x}^{3}}{3\sqrt{1+{x}^{2}}} \ y(\sqrt{3})=\frac{15\sqrt{3}}{32}=\frac{5\sqrt{3}}{2} \end{gathered}$

Solutions · 1