QJEEM21Ma131a: JEE Main 2021 Mathematics - Class 12 JEEM21Ma131a Solution: Let y = y(x) be the solution of the differential equation dy =…

Question

Verified Solution • 2025–26 Session moderate

Let y = y(x) be the solution of the differential equation dy = e^{$\alpha$ x + y} dx;

\alpha

\in

N. If y(log_e2) = log_e2 and y(0) = log_e

\left( {{1 \over 2}} \right)

, then the value of

\alpha

is equal to _____________.

Accepted Answer

\int {{e^{ - y}}} dy = \int {{e^{\alpha x}}} dx

\Rightarrow {e^{ - y}} = {{{e^{\alpha x}}} \over \alpha } + c

..... (i)

Put (x, y) = (ln2, ln2)

{{ - 1} \over 2} = {{{2^\alpha }} \over \alpha } + C

..... (ii)

Put (x, y)

\equiv

(0,

-

ln2) in (i)

- 2 = {1 \over \alpha } + C

..... (iii)

(ii)

-

(iii)

{{{2^\alpha } - 1} \over \alpha } = {3 \over 2}

\Rightarrow

\alpha

= 2 (as

\alpha

\in

N)

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