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

JEE Main · 2025 · 24 Jan 2025 · Shift 1 · Differential Equations

Let

y = y (x)

be the solution of the differential equation

(x y - 5 x^{2} 1 + x^{2}) d x + (1 + x^{2}) d y = 0

y (0) = 0

. Then

y (3)

is equal to

Answer: (A)

Divide the given equation by

(1 + x^{2})

\frac{d y}{d x} + \frac{x}{1 + x ^{2}} y = \frac{5 x ^{2}}{1 + x ^{2}} .

Integrating factor:

exp (\int \frac{x}{1 + x ^{2}} d x) = exp (\frac{1}{2} ln (1 + x^{2})) = 1 + x^{2}

\frac{d}{d x} [y 1 + x^{2}] = 1 + x^{2} \cdot \frac{5 x ^{2}}{1 + x ^{2}} = 5 x^{2} .

Integrating:

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

.

Using

y (0) = 0

0 = 0 + C \Rightarrow C = 0

, so

y = \frac{5 x ^{3}}{3 1 + x ^{2}} .

x = 3

x^{3} = 33

1 + 3 = 2

y (3) = \frac{5 \cdot 3 3}{3 \cdot 2} = \frac{5 3}{2} .

JEE Main · 2025 · 24 Jan 2025 · Shift 1 · Single correct option

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