Let y=y(x) be the solution of the differential equation x ~d y ~d x -2 y=2+3 x, x ∈(-(π)/(2), (π)/(2)) , y(0)=-(7)/(4) . Then y((π)/(6)) is equal to :

Correct answer: B. -(5)/(2) The differential equation is x (dy)/(dx) - 2y = 2 + 3 x , or (dy)/(dx) - 2 x · y = (2 + 3 x) x . This is a linear ODE with integrating factor μ(x) = e^ -2 x . Multiplying both sides and integrating: (d)/(dx)[ye^ -2 x ] = (2 + 3 x) x · e^ -2 x Computing the integral: ∫ 2 x · e^ -2 x dx = -e^ -2 x For the second term, using integration by parts: ∫ 3 x x · e^ -2 x dx = (3)/(4)e^ -2 x (-2 x - 1) Thus: ye^ -2 x = -e^ -2 x + (3)/(4)e^ -2 x (-2 x - 1) + C Simplifying: y = -(7)/(4) - (3)/(2) x + Ce^ 2 x Using y(0) = -(7)/(4) gives C = 0 . Therefore: y((π)/(6)) = -(7)/(4) - (3)/(2) · (1)/(2) = -(5)/(2)

JEE Main 202621 JanuaryEvening Shift

Exam: JEE Main
Year: 2026
Date: 21 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Differential Equations

JEE Main 2026 · 21 January · Evening Shift · Mathematics · Differential Equations

Let y=y(x) be the solution of the differential equation x ~d y ~d x...

Let

y=y(x)

be the solution of the differential equation

\sec x \frac{\mathrm{~d} y}{\mathrm{~d} x}-2 y=2+3 \sin x, 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 :

Options

A $-\frac{5}{4}$
B $-\frac{5}{2}$
C $-3 \sqrt{3}-7$
D $-3 \sqrt{2}-7$

Correct answer

B. $-\frac{5}{2}$

Step-by-step solution

The differential equation is

\sec x \frac{dy}{dx} - 2y = 2 + 3\sin x

, or

\frac{dy}{dx} - 2\cos x \cdot y = (2 + 3\sin x)\cos x

.
This is a linear ODE with integrating factor

\mu(x) = e^{-2\sin x}

.
Multiplying both sides and integrating:

\frac{d}{dx}[ye^{-2\sin x}] = (2 + 3\sin x)\cos x \cdot e^{-2\sin x}

Computing the integral:

\int 2\cos x \cdot e^{-2\sin x} dx = -e^{-2\sin x}

For the second term, using integration by parts:

\int 3\sin x \cos x \cdot e^{-2\sin x} dx = \frac{3}{4}e^{-2\sin x}(-2\sin x - 1)

Thus:

ye^{-2\sin x} = -e^{-2\sin x} + \frac{3}{4}e^{-2\sin x}(-2\sin x - 1) + C

Simplifying:

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

Using

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

gives

C = 0

.
Therefore:

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

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