The solution $\mathbf { r } ( t )$ to the differential equation $\mathbf { r } ^ { \prime } ( t ) = \sin t \mathbf { i } + \cos t \mathbf { j } + 3 \mathbf { k }$ with the condition $\mathbf { r } ( 0 ) = \mathbf { i } + \mathbf { j }$ is

A) $( \cos t + 2 ) \mathbf { i } - ( \sin t + 1 ) \mathbf { j } + 3 t \mathbf { k }$
B) $( - \cos t + 2 ) \mathbf { i } - ( \sin t + 1 ) \mathbf { j } + 3 t \mathbf { k }$
C) $( \cos t + 2 ) \mathbf { i } + ( \sin t + 1 ) \mathbf { j } + 3 t \mathbf { k }$
D) $( - \cos t + 2 ) \mathbf { i } + ( \sin t + 1 ) \mathbf { j } + 3 t \mathbf { k }$
E) $( \cos t + 2 ) \mathbf { i } + ( \sin t - 1 ) \mathbf { j } + 3 t \mathbf { k }$

Question

The solution

\mathbf { r } ( t )

to the differential equation

\mathbf { r } ^ { \prime } ( t ) = \sin t \mathbf { i } + \cos t \mathbf { j } + 3 \mathbf { k }

with the condition

\mathbf { r } ( 0 ) = \mathbf { i } + \mathbf { j }

is

A) $( \cos t + 2 ) \mathbf { i } - ( \sin t + 1 ) \mathbf { j } + 3 t \mathbf { k }$
B) $( - \cos t + 2 ) \mathbf { i } - ( \sin t + 1 ) \mathbf { j } + 3 t \mathbf { k }$
C) $( \cos t + 2 ) \mathbf { i } + ( \sin t + 1 ) \mathbf { j } + 3 t \mathbf { k }$
D) $( - \cos t + 2 ) \mathbf { i } + ( \sin t + 1 ) \mathbf { j } + 3 t \mathbf { k }$
E) $( \cos t + 2 ) \mathbf { i } + ( \sin t - 1 ) \mathbf { j } + 3 t \mathbf { k }$

Quizplus · Accepted Answer

To find the solution $\mathbf{r}(t)$, we integrate each component of $\mathbf{r}^{\prime}(t)$ with respect to $t$. Integrating $\sin t$ with respect to $t$ gives $-\cos t + C$, integrating $\cos t$ gives $\sin t + C$, and integrating $3$ gives $3t + C$. Using the initial condition $\mathbf{r}(0) = \mathbf{i} + \mathbf{j}$, we find the constants for each component. For $\mathbf{i}$, $-\cos(0) + C = 1$ gives $C = 2$, so the $\mathbf{i}$ component is $-\cos t + 2$. For $\mathbf{j}$, $\sin(0) + C = 1$ gives $C = 1$, so the $\mathbf{j}$ component is $\sin t + 1$. The $\mathbf{k}$ component is $3t$, as there's no initial condition affecting it directly. Therefore, the correct solution is $(-\cos t + 2)\mathbf{i} + (\sin t + 1)\mathbf{j} + 3t\mathbf{k}$.

The Solution r(t)\mathbf { r } ( t )r(t) To the Differential Equation

The Solution $\mathbf { r } ( t )$ To the Differential Equation