(Solved): Solve the following initial value problem for \( r \) as a function of \( t \). \[ \begin{array}{ll ...

Solve the following initial value problem for \( r \) as a function of \( t \). \[ \begin{array}{ll} \text { Differential equation: } & \frac{d^{2} \mathbf{r}}{d t^{2}}=e^{t_{i}}-4 e^{-t_{j}+18 e^{3 t} \mathbf{k}} \\ \text { Initial conditions: } & \mathbf{r}(0)=3 \mathbf{i}+\mathbf{j}+4 \mathbf{k} \\ & \left.\frac{\mathrm{dr}}{\mathrm{dt}}\right|_{\mathrm{t}=0}=-3 \mathbf{i}+6 \mathbf{j} \end{array} \] \[ \mathbf{r}(\mathrm{t})=(\quad i+1 \quad j+(\quad \mathbf{k} \]