By separation of variables,
d y d t = e y / 3 cos ( t ) e − y / 3 d y = cos ( t ) d t ∫ e − y / 3 d y = ∫ cos ( t ) d t − 3 e − y / 3 = sin ( t ) + c 1 e − y / 3 = − sin ( t ) 3 − c 1 3 e − y / 3 = − sin ( t ) 3 + c 2 , c 2 := − c 1 3 . {\displaystyle {\begin{aligned}{\frac {dy}{dt}}&=e^{y/3}\cos(t)\\e^{-y/3}\,dy&=\cos(t)\,dt\\\int e^{-y/3}\,dy&=\int \cos(t)\,dt\\-3e^{-y/3}&=\sin(t)+c_{1}\\e^{-y/3}&=-{\frac {\sin(t)}{3}}-{\frac {c_{1}}{3}}\\e^{-y/3}&=-{\frac {\sin(t)}{3}}+c_{2},\quad c_{2}:=-{\frac {c_{1}}{3}}.\end{aligned}}}
Taking the logarithm of both sides, we obtain
− y 3 = ln ( − sin ( t ) 3 + c 2 ) y = − 3 ln ( − sin ( t ) 3 + c 2 ) y = − 3 ln ( C − sin ( t ) 3 ) {\displaystyle {\begin{aligned}-{\frac {y}{3}}&=\ln \left(-{\frac {\sin(t)}{3}}+c_{2}\right)\\y&=-3\ln \left(-{\frac {\sin(t)}{3}}+c_{2}\right)\\y&=\color {blue}-3\ln \left(C-{\frac {\sin(t)}{3}}\right)\end{aligned}}}
for some constant C . {\displaystyle C.}
