Let the solution y = y(x) of the differential equation d y d x – y = 1 + 4 sin x satisfy y = 1. Then y (2) + 10 is equal to __________. [2024]

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Solution

Q.
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 __________. [2024]

Ans.
(7)

Given, $\frac{d y}{d x} - y = 1 + 4 \sin x$ , which is a linear differential equation.

$∴$ Integrating factor = $= e^{- \int d x} = e^{- x}$

Hence, solution is $y e^{- x} = \int (1 + 4 \sin x) e^{- x} d x$

$= - e^{- x} + 2 e^{- x} (- \sin x - \cos x) + C$ ... (i)

y( $π$ ) = 1 i.e., at x = $π$ , y = 1

$∴$ From (i), we get

$(1) e^{- π} = - e^{- π} + 2 e^{- π} (- \sin π - \cos π) + C$

$\Rightarrow e^{- π} = - e^{- π} + 2 e^{- π} + C \Rightarrow C = 0$

Hence, y(x) = –1 – 2(sin x + cos x)

Now, $y (\frac{π}{2}) + 10 = - 1 - 2 (\sin \frac{π}{2} + \cos \frac{π}{2}) + 10$

= –1 – 2(1) + 10 = 7.

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