Mathematics - Differential Equation Question with Solution | TestHub

$\frac{dy}{dx}-y{\log }_{e}2=2^{\sin x}(\cos x-1){\log }_{e}2$ This is linear differential equation $\text{ I.F. }=e^{-{\log }_{e}2\int dx}=e^{-x{\log }_{e}2}=2^{-x}$ then general Solution is $y{2}^{-x}=\int 2^{-x}{2}^{\sin x}(\cos x-1){\log }_{e}2dx+c$ Now let $\sin x-x=t\Rightarrow (\cos x-1)dx=dt$ $\begin{array}{rl}& \therefore y{2}^{-x}={\log }_{e}2\int 2^{t}dt+c\\ & \therefore y{2}^{-x}=2^t+c\\ & \therefore y=2^{x+t}+c{2}^x\\ & \therefore y=2^{\sin x}+c{2}^x\end{array}$