If the curve $y=f(x)$ passes through the point $(1,e)$ and satisfies the differential equation $dy=y\left(2+{\log }_{e}x\right)dx,x0$, then $f(e)$ is equal to: [JEE Main 2026, 2 Apr (Shift 1)]

Question

If the curve $y=f(x)$ passes through the point $(1,e)$ and satisfies the differential equation $dy=y\left(2+{\log }_{e}x\right)dx,x>0$, then $f(e)$ is equal to: [JEE Main 2026, 2 Apr (Shift 1)]

YESPYQ · Accepted Answer

Correct answer: c) ${e}^{2e}$. $d y=y\left(2+\log _e x\right) d x$ $\frac{d y}{y}=\left(2+\log _e x\right) d x$ $\int \frac{d y}{y}=\int\left(2+\log _e x\right) d x$ $\log _e y=2 x+x \log _e x-x+C$ $\log _e y=x+x \log _e x+C$ Given curve passes through $(1, e)$, so $x=1, y=e$ $\log _e e=1+1 \log _e 1+C$ $1=1+0+C \Rightarrow C=0$ $\log _e y=x+x \log _e x$ At $x=e, f(e)=y$ $\log _e f(e)=e+e \log _e e=e+e=2 e$ $f(e)=e^{2 e}$

If the curve \(y=f(x)\) passes through the point \((1,e)\) and satisfies the differential equation \(dy=y\left(2+{\log }…

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