JEE Main · 2026 · 02 Apr 2026 · Shift 1 · Differential Equations

If the curve

y = f (x)

passes through the point

(1, e)

and satisfies the differential equation

d y = y (2 + lo g_{e} x) d x

x > 0

, then

f (e)

is equal to :

Answer: (C)

Solution

The equation separates as

\frac{d y}{y} = (2 + ln x) d x

, so

ln y = 2 x + \int ln x d x = 2 x + (x ln x - x) = x + x ln x + C

. The point

(1, e)

gives

ln e = 1 + 0 + C

, hence

C = 0

and

ln y = x (1 + ln x)

. At

x = e

ln f (e) = e (1 + 1) = 2 e

, so

f (e) = e^{2 e}

JEE Main · 2026 · 02 Apr 2026 · Shift 1 · Single correct option

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