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JEE Main 20246 AprilEvening Shift

Exam: JEE Main
Year: 2024
Date: 6 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Differential Equations

JEE Main 2024 · 6 April · Evening Shift · Mathematics · Differential Equations

If the solution y(x) of the given differential equation (e^y+1) x...

If the solution

y(x)

of the given differential equation \left(\mathrm{e}^y+1\right) \cos x \mathrm{~d} x+\mathrm{e}^y \sin x \mathrm{~d} y=0

passes through the point \left(\frac{\pi}{2}, 0\right)

, then the value of $\mathrm{e}^{y\left(\frac{\pi}{6}\right)} is equal to_________

Correct answer

0

Step-by-step solution

\begin{aligned} & \left(\mathrm{e}^{\mathrm{y}}+1\right) \cos \mathrm{x} \mathrm{dx}+\mathrm{e}^{\mathrm{y}} \sin \mathrm{x} d \mathrm{y}=0 \\ & \Rightarrow \mathrm{d}\left(\left(\mathrm{e}^{\mathrm{y}}+1\right) \sin \mathrm{x}\right)=0 \\ & \left(\mathrm{e}^{\mathrm{y}}+1\right) \sin \mathrm{x}=\mathrm{C} \end{aligned}

<br />It passes through \left(\frac{\pi}{2}, 0\right) \Rightarrow \mathrm{c}=2

Now,

x=\frac{\pi}{6} \Rightarrow \mathrm{e}^{\mathrm{y}}=3

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