Problem 2

\text{Recognize that this is a separable differential equation.} \\ \\ \begin{aligned} \dfrac{dy}{dt} &= -y \sqrt{t} \sin(t) && \text{Subtract } y \sqrt{t} \sin(t) \text{ from both sides.} \\ dy &= -y \sqrt{t} \sin(t) dt  && \text{Multipy by } dt \text{.} \\ \dfrac{dy}{y} &= - \sqrt{t} \sin(t) dt && \text{Divide by } y \text{ on both sides.} \\ \int \dfrac{dy}{y} &= \int - \sqrt{t} \sin(t) dt && \text{Prepare to integrate.} \\  \ln |y| &= \int - \sqrt{t} \sin(t) dt \mspace{100.0mu} && \text{Integrate left side.} \\ y &= e^{\int - \sqrt{t} \sin(t) dt} && \text{Rewrite in exponential form.} \\ \\ \text{Answer: } \boxed{y = e^{\int - \sqrt{t} \sin(t) dt}} \end{aligned}

Note: The above solution would require the incomplete gamma function to integrate the right-hand side; this is beyond the assumed scope of this problem.

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