JEE Main · 2025 · 23 Jan 2025 · Shift 1 · Differential Equations

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} dx - (7 + e^{2 x}) dy = 0

, then k is equal to

Answer: (B)

Solution

The differential equation

2 (3 + y) e^{2 x} d x - (7 + e^{2 x}) d y = 0

is separable:

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

Integrating, with

u = 7 + e^{2 x}

on the right (

d u = 2 e^{2 x} d x

ln (3 + y) = ln (7 + e^{2 x}) + C ⟹ 3 + y = K (7 + e^{2 x})

Using

(0, 5)

8 = K (8) ⟹ K = 1

, so

y = 4 + e^{2 x}

x = lo g_{e} 2

k = 4 + e^{2 l o g_{e} 2} = 4 + 4 = 8

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