Answer
\[
\mathbf{r}(t)=e^{t} \mathbf{r}_{0}
\]
Work Step by Step
\[
\text { Key: } \quad \frac{d}{d t}\left(c^{t}\right)=c^{t}
\]
In 2d-space:
\[
\begin{array}{l}
\operatorname{Let} \mathrm{r}(t)=x(t) \mathrm{i}+y(t) \mathrm{j}, \quad \frac{d x}{d t}=x(t), \frac{d y}{d t}=y(t) \\
x_{0}, y(0)=y_{0} =x(0)\\
x_{0} e^{t}, y(t)=y_{0} c^{t}=x(t) \\
e^{t} \mathrm{r}_{0}=\mathrm{r}(t)
\end{array}
\]
For 3d-space, the same approach holds.
