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(A) The given differential equation is $\frac{dy}{dx} = 2x(y + 2 \sin x - 5) - 2 \cos x$.Rearranging the terms,we get $\frac{dy}{dx} - 2xy = 2x(2 \sin

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$2 e^{\pi^{2}}+5$

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(A) The given differential equation is $\frac{dy}{dx} = 2x(y + 2 \sin x - 5) - 2 \cos x$.Rearranging the terms,we get $\frac{dy}{dx} - 2xy = 2x(2 \sin x - 5) - 2 \cos x$.This is a linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$,where $P(x) = -2x$ and $Q(x) = 4x \sin x - 10x - 2 \cos x$.The integrating factor $IF = e^{\int P(x) dx} = e^{\int -2x dx} = e^{-x^{2}}$.Multiplying both sides by $IF$,we get $e^{-x^{2}} \frac{dy}{dx} - 2x e^{-x^{2}} y = e^{-x^{2}}(4x \sin x - 10x - 2 \cos x)$.This simplifies to $\frac{d}{dx}(y \cdot e^{-x^{2}}) = e^{-x^{2}}(4x \sin x - 2 \cos x) - 10x e^{-x^{2}}$.Note that $\frac{d}{dx}(e^{-x^{2}}(5 - 2 \sin x)) = e^{-x^{2}}(-2 \cos x) + (5 - 2 \sin x)(-2x e^{-x^{2}}) = -2 \cos x e^{-x^{2}} - 10x e^{-x^{2}} + 4x \sin x e^{-x^{2}}$.Thus,$y \cdot e^{-x^{2}} = e^{-x^{2}}(5 - 2 \sin x) + C$.Dividing by $e^{-x^{2}}$,we get $y = 5 - 2 \sin x + C e^{x^{2}}$.Given $y(0) = 7$,we have $7 = 5 - 2 \sin(0) + C e^{0} \Rightarrow 7 = 5 + C \Rightarrow C = 2$.So,$y(x) = 5 - 2 \sin x + 2 e^{x^{2}}$.At $x = \pi$,$y(\pi) = 5 - 2 \sin(\pi) + 2 e^{\pi^{2}} = 5 - 0 + 2 e^{\pi^{2}} = 2 e^{\pi^{2}} + 5$.

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2(y+2 \sin x-5)x-2 \cos x$ such that $y(0)=7$. Then $y(\pi)$ is equal to :

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