Which of the following is the general solution to the differential equation and boundary condition shown below? 

$$dy/dt+10y=0; y(0)=3$$

This equation is a first order homogenous differential equation. It takes the form of:

$y^{\text{'}}+ay=0$

The general solution to a first order homogenous is:

$y=C{e}^{-at}$

Solving for the general solution of the equation in our problem:

$$\begin{gathered} \frac{dy}{dt}+10y=0 \\ {y}^{\text{'}}+10y=0 \\ y=C{e}^{-10t} \end{gathered}$$

We can solve for our constant, C, using the boundary condition we are given:

$$\begin{gathered} y=C{e}^{-10t} \\ 3 = C{e}^{-10\left(0\right)} \\ 3 = C\left(e^0\right) \\ 3 = C\left(1\right) \\ 3 = C \end{gathered}$$

Therefore, our final form of our general solution is:

$\mathit{y}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{3}{\mathit{e}}^{\mathbf{-}\mathbf{10}\mathbf{t}}$