Problem 1

\text{Recognize that this is a separable differential equation.} \\ \\ \begin{aligned} \dfrac{dy}{dt} &= -y \cos(t) && \text{Subtract } y \cos(t) \text{ from both sides.} \\ dy &= -y \cos(t) dt  && \text{Multipy by } dt \text{.} \\ \dfrac{dy}{y} &= \cos(t) dt && \text{Divide by } y \text{ on both sides.} \\ \int \dfrac{dy}{y} &= \int \cos(t) dt && \text{Prepare to integrate.} \\ \ln |y| &= \sin(t) + C \mspace{100.0mu} && \text{Integrate both sides.} \\ y &= e^{\sin(t) + C} && \text{Rewrite in exponential form.} \\ y &= e^{\sin(t)} \cdot e^{C} && \text{Rewrite as the multiplication of two common bases.} \\ y &= Ce^{\sin(t)} && \text{Let } e^{C}=C \text{.} \\ \\ \text{Answer: } \boxed{y = Ce^{\sin(t)}} \end{aligned}