JEE Main 2025 Jan 24 Shift 1, Mathematics Q9: Methods of Solving First Order, First Degree Differential Equation

Rearrange into standard linear form $\frac{dy}{dx}+P(x)y=Q(x)$:

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

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

Integrating factor:

$\mu =\exp \int \frac{x}{1+x^2}dx=\exp \left(\frac{1}{2}\ln (1+x^2)\right)=\sqrt{1+x^2}$

Multiply through:

$\frac{d}{dx}\left(y\sqrt{1+x^2}\right)=\frac{5x^2}{\sqrt{1+x^2}}\cdot \sqrt{1+x^2}=5x^2$

Integrate:

$y\sqrt{1+x^2}=\frac{5x^3}{3}+C$

$y(0)=0\Rightarrow C=0$, so $y=\frac{5x^3}{3\sqrt{1+x^2}}$.

At $x=\sqrt{3}$: $y=\frac{5\cdot 3\sqrt{3}}{3\cdot 2}=\frac{5\sqrt{3}}{2}$.

$\therefore$ Option (1).