QJEEM20Ma4c11: JEE Main 2020 Mathematics - Class 12 JEEM20Ma4c11 Solution: If y = y(x) is the solution of the differential equation…

Question

Verified Solution • 2025–26 Session moderate

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

equation

{{5 + {e^x}} \over {2 + y}}.{{dy} \over {dx}} + {e^x} = 0

satisfying y(0) = 1, then a value of y(log_e13) is :

Accepted Answer

{{5 + {e^x}} \over {2 + y}}.{{dy} \over {dx}} + {e^x} = 0

\Rightarrow

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

Integrating both sides,

\Rightarrow

\int {{{dy} \over {2 + y}}} = \int {{{ - {e^x}} \over {{e^x} + 5}}} dx

\Rightarrow

ln (y + 2) = – ln(e^x + 5) + k

\Rightarrow

(y + 2) (e^x + 5) = C

\because

y(0) = 1

\Rightarrow

C = 18

\therefore

y + 2 =

{{{18} \over {{e^x} + 5}}}

At x = log_e13

y + 2 =

{{{18} \over {13 + 5}}}

= 1

\Rightarrow

y = -1

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