Let a curve $y=f(x)$ pass through the point $\left(2,\left(\log _{e} 2\right)^{2}\right)$ and have slope $\frac{2 y}{x \log _{e} x}$ for all positive real value of $x$. Then the value of $f(e)$ is equal to ____

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Differential Equations

Let a curve

y = f (x)

pass through the point

(2, (lo g_{e} 2)^{2})

and have slope

\frac{2 y}{x l o g _{e} x}

for all positive real value of

x

. Then the value of

f (e)

is equal to ____

Answer: 1

The differential equation is:

\frac{d y}{d x} = \frac{2 y}{x l n x}

Separating variables:

\frac{d y}{y} = \frac{2 d x}{x l n x}

Integrating both sides:

ln ∣ y ∣ = 2 ln ∣ ln x ∣ + C

ln ∣ y ∣ = ln (ln x)^{2} + C

y = A (ln x)^{2}

where

A = e^{C}

Using the initial condition

y (2) = (ln 2)^{2}

(ln 2)^{2} = A (ln 2)^{2}

A = 1

Therefore,

y = (ln x)^{2}

.

At

x = e

f (e) = (ln e)^{2} = 1^{2} = 1

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Integer answer

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