QJEEM21Ma137e: JEE Main 2021 Mathematics - Class 12 JEEM21Ma137e Solution: If , y(0) = 0, then for y = 1, the value of x lies in the…

Question

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If

{{dy} \over {dx}} = {{{2^x}y + {2^y}{{.2}^x}} \over {{2^x} + {2^{x + y}}{{\log }_e}2}}

, y(0) = 0, then for y = 1, the value of x lies in the interval :

Accepted Answer

{{dy} \over {dx}} = {{{2^x}(y + {2^y})} \over {{2^x}(1 + {2^y}\ln 2)}}

\Rightarrow \int {{{(1 + {2^y})\ln 2} \over {(y + {2^y})}}dy = \int {dx} }

\Rightarrow \ln \left| {y + {2^y}} \right| = x + c

x = 0; y = 0

\Rightarrow

c = 0

\Rightarrow x = \ln \left| {y + {2^y}} \right|

\Rightarrow

at y = 1, x = ln3

\because

3 \in (e,{e^2}) \Rightarrow x \in (1,2)

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