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Q.
If $\frac{dy}{dx}+y\,cos\,x=\frac{1}{2}\,sin\,2x$, $y\left(0\right)=1$, then $y\left(\frac{\pi}{2}\right)=$
Differential Equations
Solution:
It is linear differential equation with $I$.$F$.
$= \text{exp} \int \,cos\,x\,dx=e^{sin\,x}$.
The solution is, $ye^{sinx}=\int sin\,x\,cos\,x\,e^{sin\,x}\,dx$
$\Rightarrow ye^{sinx}=\left(sinx-1\right)e^{sinx}+c$
$\Rightarrow y=sin\,x-1+ce^{-sin\,x}$
$y\left(0\right)=1$
$\Rightarrow c=2$
$\Rightarrow y=sin\,x-1+2e^{-sin\,x}$
$\therefore y\left(\frac{\pi}{2}\right)=\frac{2}{e}$.