7. Find r(t) if r'(t) = 6e³ti r(0) = 5i + 3k. + costj + sec²t k and
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 7: Find r(t)**
Given:
\[ \mathbf{r}'(t) = 6e^{3t} \mathbf{i} + \cos(t) \mathbf{j} + \sec^2(t) \mathbf{k} \]
with the initial condition:
\[ \mathbf{r}(0) = 5\mathbf{i} + 3\mathbf{k} \]
To find the position vector \(\mathbf{r}(t)\), we need to integrate \(\mathbf{r}'(t)\).
1. **Integrate the \( \mathbf{i} \) component:**
\[ \int 6e^{3t} \, dt = 2e^{3t} + C_1 \]
2. **Integrate the \( \mathbf{j} \) component:**
\[ \int \cos(t) \, dt = \sin(t) + C_2 \]
3. **Integrate the \( \mathbf{k} \) component:**
\[ \int \sec^2(t) \, dt = \tan(t) + C_3 \]
Thus, the general form of \(\mathbf{r}(t)\) before applying initial conditions is:
\[ \mathbf{r}(t) = (2e^{3t} + C_1) \mathbf{i} + (\sin(t) + C_2) \mathbf{j} + (\tan(t) + C_3) \mathbf{k} \]
Using the initial condition \(\mathbf{r}(0) = 5\mathbf{i} + 3\mathbf{k}\):
- For the \( \mathbf{i} \) component:
\[ 2e^{0} + C_1 = 5 \]
\[ 2 + C_1 = 5 \]
\[ C_1 = 3 \]
- For the \( \mathbf{j} \) component:
\[ \sin(0) + C_2 = 0 \]
\[ 0 + C_2 = 0 \]
\[ C_2 = 0 \]
- For the \( \mathbf{k} \) component:
\[ \tan(0) + C_3 = 3 \]
\[ 0 + C_3 = 3 \]
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55815b3c-2575-4201-a9dc-5a4b41a14c1d%2Fbdf5571f-8dd3-48cc-9cf6-0b4653145800%2Fi9dp6n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 7: Find r(t)**
Given:
\[ \mathbf{r}'(t) = 6e^{3t} \mathbf{i} + \cos(t) \mathbf{j} + \sec^2(t) \mathbf{k} \]
with the initial condition:
\[ \mathbf{r}(0) = 5\mathbf{i} + 3\mathbf{k} \]
To find the position vector \(\mathbf{r}(t)\), we need to integrate \(\mathbf{r}'(t)\).
1. **Integrate the \( \mathbf{i} \) component:**
\[ \int 6e^{3t} \, dt = 2e^{3t} + C_1 \]
2. **Integrate the \( \mathbf{j} \) component:**
\[ \int \cos(t) \, dt = \sin(t) + C_2 \]
3. **Integrate the \( \mathbf{k} \) component:**
\[ \int \sec^2(t) \, dt = \tan(t) + C_3 \]
Thus, the general form of \(\mathbf{r}(t)\) before applying initial conditions is:
\[ \mathbf{r}(t) = (2e^{3t} + C_1) \mathbf{i} + (\sin(t) + C_2) \mathbf{j} + (\tan(t) + C_3) \mathbf{k} \]
Using the initial condition \(\mathbf{r}(0) = 5\mathbf{i} + 3\mathbf{k}\):
- For the \( \mathbf{i} \) component:
\[ 2e^{0} + C_1 = 5 \]
\[ 2 + C_1 = 5 \]
\[ C_1 = 3 \]
- For the \( \mathbf{j} \) component:
\[ \sin(0) + C_2 = 0 \]
\[ 0 + C_2 = 0 \]
\[ C_2 = 0 \]
- For the \( \mathbf{k} \) component:
\[ \tan(0) + C_3 = 3 \]
\[ 0 + C_3 = 3 \]
\[
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