JEE Main 2025 Jan 23 Shift 1, Mathematics Q5: Methods of Solving First Order, First Degree Differential Equation

JEE Main2025Jan 23, Shift 1Mathematics

Let a curve $y = f (x)$ pass through the points $(0, 5)$ and $(lo g_{e} 2, k)$ . If the curve satisfies the differential equation $2 (3 + y) e^{2 x} d x - (7 + e^{2 x}) d y = 0$ , then $k$ is equal to

A
$16$
B
$8$
C
$32$
D
$4$

Solution

The equation is separable. Rearranging:

$\frac{d y}{3 + y} = \frac{2 e ^{2 x}}{7 + e ^{2 x}} d x$ .

Both sides integrate to logs:

$ln ∣3 + y ∣ = ln ∣7 + e^{2 x} ∣ + C_{1}$ , so $3 + y = C (7 + e^{2 x})$ .

Apply the initial condition $(0, 5)$ : $3 + 5 = C (7 + 1) \Rightarrow C = 1$ .

Hence $y = 4 + e^{2 x}$ .

At $x = ln 2$ , $e^{2 x} = 4$ , so $y = 8$ .

$∴ k = 8$ .

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JEE Main 2025 Jan 23 Shift 1, Mathematics Q5: Methods of Solving First Order, First Degree Differential Equation

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