If the solution $y(x)$ of the given differential equation $\left(e^{y}+1\right) \cos x d x+e^{y} \sin x d y=0$ passes through the point $\left(\frac{\pi}{2}, 0\right)$ then the value of $e^{y\left(\frac{\pi}{6}\right)}$ is equal to ______.

JEE Main · 2024 · 06 Apr 2024 · Shift 2 · Differential Equations

If the solution

y (x)

of the given differential equation

(e^{y} + 1) cos x d x + e^{y} sin x d y = 0

passes through the point

(\frac{π}{2}, 0)

then the value of

e^{y (\frac{π}{6})}

is equal to ______.

Answer: 3

(e^{y} + 1) cos x d x + e^{y} sin x d y = 0

is exact: it is

d [(e^{y} + 1) sin x] = 0

, so

(e^{y} + 1) sin x = C

. At

(\frac{π}{2}, 0)

(1 + 1) (1) = 2 = C

. At

x = \frac{π}{6}

(e^{y} + 1) \cdot \frac{1}{2} = 2 \Rightarrow e^{y} + 1 = 4 \Rightarrow e^{y (π /6)} = 3

JEE Main · 2024 · 06 Apr 2024 · Shift 2 · Integer answer

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