If y=y(x) is the solution of the differential equation [d y~d x+2 y= (2 x), y(0)=(3)/(4)],…

Question

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

Accepted Answer

$$\begin{aligned}
& \frac{d y}{d x}+2 y=\sin 2 x \
& \text { IF }=e^{2 d x}=e^{2 x} \
& y \cdot e^{2 x}=\int e^{2 x} \sin 2 x d x+c \
& =\frac{e^{2 x}}{8}(2 \sin 2 x-2 \cos 2 x)+c \
& y(0)=\frac{3}{4} \
& \frac{3}{4}=\frac{1}{8}(-2)+c \Rightarrow c=1 \
& \text { Put } x=\frac{\pi}{8} \
& y=\frac{1}{8} \times 2\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\right)+e^{-\pi / 4} \
& y=e^{-\pi / 4}
\end{aligned}$$

Solutions · 1