Equação diferencial

Creio que seja isso.

[latex]\\\mathrm{Seja\ o\ fator\ integrante:\ \nu (x)=e^{\int cos(x)dx}=e^{sin(x)}}\\\\\mathrm{\frac{dy}{dx}+ycos(x)=4sin(2x)\to e^{sin(x)}\frac{dy}{dx}+ye^{sin(x)}cos(x)=4e^{sin(x)}sin(2x)}\\\\\mathrm{e^{sin(x)}\frac{dy}{dx}+y\frac{d}{dx}[e^{sin(x)}]=4e^{sin(x)}sin(2x)\to \frac{d}{dx}\left [ ye^{sin(x)} \right ]=4e^{sin(x)}sin(2x)}\\\\\mathrm{\int \frac{d}{dx}\left [ ye^{sin(x)} \right ]=\int 4e^{sin(x)}sin(2x)dx\to ye^{sin(x)}=8e^{sin(x)}sin(x)-8e^{sin(x)}+C}\\\\\mathrm{y=8sin(x)-8+Ce^{-sin(x)}\ \therefore \ y(0)=15\to C=23\ \therefore \ y=8sin(x)+23e^{-sin(x)}-8}\\\\\mathrm{\therefore \ y(5)=8sin(5)+23e^{-sin(5)}-8}[/latex]