Let y = y(x) be the solution of the differential equation sec x ? d y d x - 2 y = 2 + 3 sin x , x ( -2 , ?2 ) , y ( 0 ) = - 7 4 . Then y ( 6 ) is equal to: [2026]

Question

Let $y = y (x)$ be the solution of the differential equation $\sec x \frac{d y}{d x} - 2 y = 2 + 3 \sin x, x \in (- \frac{π}{2}, \frac{π}{2}),$ $y (0) = - \frac{7}{4} .$ Then $y (\frac{π}{6})$ is equal to: [2026]

Smart Achievers · Accepted Answer

(1)

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

$I.F. = e^{- 2 \sin x}$

$e^{- 2 \sin x} y = \int e^{- 2 \sin x} (3 \sin x \cos x + 2 \cos x) d x$

$y . e^{- 2 \sin x} = e^{- 2 \sin x} (- \frac{3}{2} \sin x - \frac{7}{4}) + C$

$\Rightarrow y = - \frac{3}{2} \sin x - \frac{7}{4} + C e^{2 \sin x}$

$∵$ $y (0) = - \frac{7}{4} \Rightarrow C = 0$

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

Solution

Q.
Let $y = y (x)$ be the solution of the differential equation $\sec x \frac{d y}{d x} - 2 y = 2 + 3 \sin x, x \in (- \frac{π}{2}, \frac{π}{2}),$ $y (0) = - \frac{7}{4} .$ Then $y (\frac{π}{6})$ is equal to: [2026]

1 $- \frac{5}{2}$

2 $- \frac{5}{4}$

3 $- 3 \sqrt{3} - 7$

4 $- 3 \sqrt{2} - 7$

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let y=y(x) be the solution of the differential equation secxdydx-2y=2+3sinx, x∈(-π2,π2), y(0)=-74.Then y(π6) is equal to: [2026]

1 -52

2 -54

3 -33-7

4 -32-7