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JEE Main 20244 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
4 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Differential Equations

JEE Main 2024 · 4 April · Morning Shift · Mathematics · Differential Equations

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

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

Correct answer

0

Step-by-step solution

ye−x=∫(e−x+4e−xsin⁡x)dxye−x=−e−x−2(e−xsin⁡xe−xcos⁡x)+Cy=−1−2(sin⁡x+cos⁡x)+cex∵y(π)=1⇒c=0y(π/2)=−1−2=−3 Ans =10−3=7\begin{aligned} & y e^{-x}=\int\left(e^{-x}+4 e^{-x} \sin x\right) d x \\ & y e^{-x}=-e^{-x}-2\left(e^{-x} \sin x e^{-x} \cos x\right)+C \\ & y=-1-2(\sin x+\cos x)+c e^x \\ & \because y(\pi)=1 \Rightarrow c=0 \\ & y(\pi / 2)=-1-2=-3 \\ & \text{ Ans }=10-3=7\end{aligned}
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