Q. 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 $y(\sqrt3)$ is equal to
A) $\sqrt{\frac{15}2}$
B) $\frac{5\sqrt3}2$
C) $2\sqrt2$
D) $\sqrt{\frac{14}3}$
Explanation
[Solution by Dr. Abubacker] $\frac{dy}{dx}+\frac{x}{1+x^2}y=\frac{5x^2}{\sqrt{1+x^2}}$. The integrating factor is $\sqrt{1+x^2}$, so $y\sqrt{1+x^2}=\int5x^2dx=\frac{5x^3}3+C$. $y(0)=0\Rightarrow C=0$. $y(\sqrt3)=\frac{5\cdot3\sqrt3/3}{2}=\frac{5\sqrt3}2$. Answer (2).
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.