Let the solution $y=y(x)$ of the differential equation $\frac{dy}{dx}-y=1+4\sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+10$ is equal to ____.

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Differential Equations

Let the solution

y = y (x)

of the differential equation

\frac{d y}{d x} - y = 1 + 4 sin x

satisfy

y (π) = 1

. Then

y (\frac{π}{2}) + 10

is equal to ____.

Answer: 7

Linear ODE

\frac{d y}{d x} - y = 1 + 4 sin x

; integrating factor

e^{- x}

y e^{- x} = \int e^{- x} (1 + 4 sin x) d x = - e^{- x} - 2 e^{- x} (sin x + cos x) + C .

y = - 1 - 2 (sin x + cos x) + C e^{x}

y (π) = - 1 - 2 (0 - 1) + C e^{π} = 1 + C e^{π} = 1 \Rightarrow C = 0

y (\frac{π}{2}) = - 1 - 2 (1 + 0) = - 3 \Rightarrow y (\frac{π}{2}) + 10 = 7.

Answer: 7.

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