If (2+sin x) frac{dy}{dx} + (y+1) cos x = 0 a | vedclass

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(D) Given the differential equation: $(2+\sin x) \frac{dy}{dx} + (y+1) \cos x = 0$ Rearranging the terms to separate the variables: $\frac{dy}{y+1} =

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$\frac{1}{3}$

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(D) Given the differential equation: $(2+\sin x) \frac{dy}{dx} + (y+1) \cos x = 0$ Rearranging the terms to separate the variables: $\frac{dy}{y+1} = -\frac{\cos x}{2+\sin x} dx$ Integrating both sides: $\int \frac{dy}{y+1} = -\int \frac{\cos x}{2+\sin x} dx$ $\ln(y+1) = -\ln(2+\sin x) + C$ $\ln(y+1) + \ln(2+\sin x) = C$ $\ln((y+1)(2+\sin x)) = C$ $(y+1)(2+\sin x) = e^C = K$ Given $y(0) = 1$,substitute $x=0$ and $y=1$: $(1+1)(2+\sin 0) = K \Rightarrow 2(2+0) = K \Rightarrow K = 4$ So,$(y+1)(2+\sin x) = 4$ For $x = \frac{\pi}{2}$: $(y+1)(2+\sin \frac{\pi}{2}) = 4$ $(y+1)(2+1) = 4$ $3(y+1) = 4$ $y+1 = \frac{4}{3}$ $y = \frac{4}{3} - 1 = \frac{1}{3}$

If $(2+\sin x) \frac{dy}{dx} + (y+1) \cos x = 0$ and $y(0) = 1$,then $y\left(\frac{\pi}{2}\right)$ is

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