QJEEM21Ma04a4: JEE Main 2021 Mathematics - Class 12 JEEM21Ma04a4 Solution: If , y(0) = 1, then y(1) is equal to :

Question

Verified Solution • 2025–26 Session moderate

If

{{dy} \over {dx}} = {{{2^{x + y}} - {2^x}} \over {{2^y}}}

, y(0) = 1, then y(1) is equal to :

Accepted Answer

{{dy} \over {dx}} = {{{2^{x + y}} - {2^x}} \over {{2^y}}}

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

\int {{{{2^y}} \over {{2^y} - 1}}dy = \int {{2^x}\,dx} }

{{\ln ({2^y} - 1)} \over {\ln 2}} = {{{2^x}} \over {\ln 2}} + C

\Rightarrow {\log _2}({2^y} - 1) = {2^x}{\log _2}e + C

\because

y(0) = 1 \Rightarrow 0 = {\log _2}e + C

C = - {\log _2}e

\Rightarrow {\log _2}({2^y} - 1) = ({2^x} - 1){\log _2}e

put x = 1,

{\log _2}({2^y} - 1) = {\log _2}e

2^y = e + 1

y = log₂(e + 1) Ans.

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