QJEEM22Ma3791: JEE Main 2022 Mathematics - Class 12 JEEM22Ma3791 Solution: If x = x(y) is the solution of the differential equation ; then …

Question

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If x = x(y) is the solution of the differential equation

$y{{dx} \over {dy}} = 2x + {y^3}(y + 1){e^y},\,x(1) = 0$ ; then x(e) is equal to :

Accepted Answer

\frac{d x}{d y}-\frac{2 x}{y}=y^{2}(y+1) e^{y}

\text { If }=e^{\int-\frac{2}{y} d y}=e^{-2 \ln y}=\frac{1}{y^{2}}

Solution is given by

\begin{aligned}&amp;x \cdot \frac{1}{y^{2}}=\int y^{2}(y+1) e^{y} \cdot \frac{1}{y^{2}} d y \\\\\Rightarrow &amp; \frac{x}{y^{2}}=\int(y+1) e^{y} d y \\\\\Rightarrow &amp; \frac{x}{y^{2}}=y e^{y}+c\end{aligned}

\Rightarrow x=y^{2}\left(y e^{y}+c\right)

at,

y=1, x=0

\Rightarrow 0=1\left(1 \cdot e^{1}+c\right) \Rightarrow c=-e

at

y=e

,

x=e^{2}\left(e . e^{e}-e\right)

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