Let $y=y(x)$ be the solution of the differential equation $\sec x \frac{\mathrm{dy}}{\mathrm{dx}}-2 \mathrm{y}=2+3 \sin \mathrm{x}, \mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, $y(0)=-\frac{7}{4}$. Then $y\left(\frac{\pi}{6}\right)$ is equal to:

JEE Main · 2026 · 21 Jan 2026 · Shift 2 · Differential Equations

Let

y = y (x)

be the solution of the differential equation

sec x \frac{dy}{dx} - 2 y = 2 + 3 sin x, x \in (- \frac{π}{2}, \frac{π}{2})

y (0) = - \frac{7}{4}

. Then

y (\frac{π}{6})

is equal to:

Answer: (A)

Rewrite as

\frac{d y}{d x} - 2 y cos x = 2 cos x + 3 sin x cos x

Integrating factor:

e^{- 2 s i n x}

Multiplying and integrating:

y \cdot e^{- 2 s i n x} = e^{- 2 s i n x} (- \frac{3}{2} sin x - \frac{7}{4}) + C

y = - \frac{3}{2} sin x - \frac{7}{4} + C e^{2 s i n x}

Using

y (0) = - \frac{7}{4}

C = 0

Therefore

y = - \frac{3}{2} sin x - \frac{7}{4}

y (\frac{π}{6}) = - \frac{3}{2} \cdot \frac{1}{2} - \frac{7}{4} = - \frac{3}{4} - \frac{7}{4} = - \frac{5}{2}

JEE Main · 2026 · 21 Jan 2026 · Shift 2 · Single correct option

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