Question

Solve the initial value problems in Exercises $11-20$ for $\mathbf{r}$ as a vector function of $t .$ Differential equation: $$\quad \frac{d^{2} \mathbf{r}}{d t^{2}}=e^{t} \mathbf{i}-e^{-t} \mathbf{j}+4 e^{2} \mathbf{k}$$ Initial conditions: $$\mathbf{r}(0)=3 \mathbf{i}+\mathbf{j}+2 \mathbf{k} and \left.\frac{d \mathbf{r}}{d t}\right|_{t=0}=-\mathbf{i}+4 \mathbf{j}$$

          Solve the initial value problems in Exercises $11-20$ for $\mathbf{r}$ as a vector function of $t .$

Differential equation: $$\quad \frac{d^{2} \mathbf{r}}{d t^{2}}=e^{t} \mathbf{i}-e^{-t} \mathbf{j}+4 e^{2} \mathbf{k}$$
Initial conditions: $$\mathbf{r}(0)=3 \mathbf{i}+\mathbf{j}+2 \mathbf{k} and
\left.\frac{d \mathbf{r}}{d t}\right|_{t=0}=-\mathbf{i}+4 \mathbf{j}$$

Added by Jeremiah A.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Brent Burkett

Step 1

To do this, we can use the method of undetermined coefficients. We assume that the solution has the form $\mathbf{r}(t)=\mathbf{A}e^t+\mathbf{B}e^{-t}+\mathbf{C}e^{2t}$, where $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ are constants to be determined. Taking the Show more…

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Solve the initial value problems in Exercises $11-20$ for $\mathbf{r}$ as a vector function of $t .$ Differential equation: $$\quad \frac{d^{2} \mathbf{r}}{d t^{2}}=e^{t} \mathbf{i}-e^{-t} \mathbf{j}+4 e^{2} \mathbf{k}$$ Initial conditions: $$\mathbf{r}(0)=3 \mathbf{i}+\mathbf{j}+2 \mathbf{k} and \left.\frac{d \mathbf{r}}{d t}\right|_{t=0}=-\mathbf{i}+4 \mathbf{j}$$