Question

$\frac{dy}{dx}=y\sin 2x,y\left(0\right)=1$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \frac{dy}{dx}=y\sin 2x,y\left(0\right)=1 \\ \Rightarrow \frac{1}{y}dy=\sin 2xdx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int \frac{1}{y}dy=\int \sin 2xdx \\ \Rightarrow \log \left|y\right|=-\frac{\cos 2x}{2}+\mathrm{C}.....\left(1\right) \\ \mathrm{Given}:x=0,y=1. \\ \mathrm{Substituting}\mathrm{the}\mathrm{values}\mathrm{of}x\mathrm{and}y\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ \log \left|1\right|=-\frac{1}{2}+C \\ \Rightarrow C=\frac{1}{2} \\ \mathrm{Substituting}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{C}\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \end{gathered}$$
$$\begin{gathered} \log \left|y\right|=-\frac{\cos 2x}{2}+\frac{1}{2} \\ \Rightarrow \log \left|y\right|=\frac{1-\cos 2x}{2} \\ \Rightarrow \log \left|y\right|=\sin {}^{2}x \\ \Rightarrow y=e^{{\sin }^{2}x} \\ \mathrm{Hence},y=e^{{\sin }^{2}x}\mathrm{is}\mathrm{the}\mathrm{required}\mathrm{solution}. \end{gathered}$$