If y=y(x) is the solution of the differential equation (dy)/(dx)+2y= (2x),y(0)=(3)/(4), th…

Question

If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+2y=\sin (2x),y(0)=\frac{3}{4}$, then $y(\frac{\pi }{8})$ is equal to :

Accepted Answer

$\begin{gathered} \frac{dy}{dx}+2y=\sin 2x \ ={e}^{2dx}={e}^{2x} \ y\cdot {e}^{2x}=\int {e}^{2x}\sin 2xdx+c \ =\frac{{e}^{2x}}{8}(2\sin 2x-2\cos 2x)+c \ y(0)=\frac{3}{4} \ \frac{3}{4}=\frac{1}{8}(-2)+c\Rightarrow c=1 \ x=\frac{\pi }{8} \ y=\frac{1}{8}	imes 2(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}})+{e}^{-\pi /4} \ y={e}^{-\pi /4} \end{gathered}$

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