Solve frac{dphi_{B}}{dt}=d/dt(2+e^-t/5sin6pit)

$$\frac { d \phi _ { B } } { d t } = \frac { d } { d t } ( 2 + e ^ { - t / 5 } \sin 6 \pi t )$$

Solve for ϕ_B (complex solution)

$$ϕ_{B}\in \mathrm{C}$$

$$-3\pi e^{6\pi it-\frac{t}{5}}-3\pi e^{-6\pi it-\frac{t}{5}}-\frac{ie^{6\pi it-\frac{t}{5}}}{10}+\frac{ie^{-6\pi it-\frac{t}{5}}}{10}=0$$

Solve for t

$$t=\frac{n_{1}+\frac{\arcsin(\frac{30\pi }{\sqrt{900\pi ^{2}+1}})}{\pi }}{6}$$

$$n_{1}\in \mathrm{Z}$$

Solve for ϕ_B

$$ϕ_{B}\in \mathrm{R}$$

$$\exists n_{1}\in \mathrm{Z}\text{ : }t=\frac{n_{1}+\frac{\arcsin(\frac{30\pi }{\sqrt{900\pi ^{2}+1}})}{\pi }}{6}$$