Mathematics - Differential Equation Question with Solution | TestHub

Given differential equation is $$\frac{(2+\sin x}{(1+y)}\cdot \frac{dy}{dx}=\cos x$$ which can be rewritten as $$\frac{dy}{1+y}=\frac{\cos x}{2+\sin x}dx$$ Integrate both the sides, we get $$\begin{array}{rl}& \int \frac{dy}{1+y}=\int \frac{\cos xdx}{2+\sin x}\\ & \Rightarrow \log (1+y)=\log (2+\sin x)+\log C\\ & \Rightarrow 1+y=C(2+\sin x)\\ & \text{ Given }y(0)=2\\ & \Rightarrow 1+2=C[2+\sin 0]\Rightarrow C=\frac{3}{2}\end{array}$$ Now, $y\left(\frac{\pi }{2}\right)$ can be found as $$1+y=\frac{3}{2}\left(2+\sin \frac{\pi }{2}\right)\Rightarrow 1+y=\frac{9}{2}$$ $$\Rightarrow y=\frac{7}{2}$$ Hence, $y\left(\frac{\pi }{2}\right)=\frac{7}{2}$