Let the solution y = y ( x ) of the differential equation d y/d x-y=1+4 sin x satisfy y(π)=1 . Then y (π/2 )+10 is equal to

Mathematics · Differential Equations

JEE Main 2024 — 4 April, Shift 1 — Question 23

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

Answer: 7

Numerical answer — enter this value.

Step-by-step solution

$\mathrm{ye}^{-\mathrm{x}}=\int\left(\mathrm{e}^{-\mathrm{x}}+4 \mathrm{e}^{-\mathrm{x}} \sin \mathrm{x}\right) \mathrm{dx}$

$\mathrm{ye}^{-\mathrm{x}}=-\mathrm{e}^{-\mathrm{x}}-2\left(\mathrm{e}^{-\mathrm{x}} \sin \mathrm{x} \mathrm{e}^{-\mathrm{x}} \cos \mathrm{x}\right)+\mathrm{C}$

$y=-1-2(\sin x+\cos x)+c e^{x}$

$\because \mathrm{y}(\pi)=1 \Rightarrow \mathrm{c}=0$

$y(\pi / 2)=-1-2=-3$

Ans $=10-3=7$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 4 April, Shift 1
Subject: Mathematics
Chapter: Differential Equations
Topic: Methods of solving a First Order,First Degree Differential

If the solution y=y(x) of the differential equation (x^4+2 x^3+3 x^2+2 x+2 ) d y- (2 x^2+2 x+3 ) d x=0 satisfies y(-1)=-pi/4 , then y(0) is…

Answer: 7

Step-by-step solution

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