2. Solve the initial value problem z""+x" - 4x' - 4x = 0, x(0) = 1, x'(0) = 0, x" (0) = -1. Hint: Guess one eigenvalue.
2. Solve the initial value problem z""+x" - 4x' - 4x = 0, x(0) = 1, x'(0) = 0, x" (0) = -1. Hint: Guess one eigenvalue.
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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Please do number 2
![### Exercises
1. **Find the general solution of the following differential equations.**
a) \( x''' + x' = 0. \)
b) \( x''' + x' = 1. \)
c) \( x'''' + x'' = 0. \)
d) \( x''' - x' - 8x = 0. \)
e) \( x''' + x'' = 2e^t + 3t^2. \)
f) \( x''' - 8x = 0. \)
2. **Solve the initial value problem** \( x'' + 4x' - 4x = 0, x(0) = 1, x'(0) = 0, x''(0) = -1. \) *Hint: Guess one eigenvalue.*
3. **Write a linear, fourth-order differential equation whose general solution is:**
\[
x(t) = c_1 + c_2 t + e^{5t} (c_4 \cos 2t + c_5 \sin 5t).
\]
4. **What is the general solution of a fourth-order differential equation if the four eigenvalues are \( \lambda = 3 \pm i, 3 \pm i \)? What is the differential equation?**
---
### 2.6 Steady-State Heat Conduction*
Let us consider the following problem in steady-state heat conduction. A cylindrical, uniform, metallic bar of length \( L \) and cross-sectional area \( A \) is insulated on its lateral side. We assume the left face at \( x = 0 \) is maintained at \( T_0 \) degrees and that the right face at \( x = L \) is held at \( T_L \) degrees. What is the temperature distribution \( u = u(x) \) in the bar after it comes to equilibrium? Here \( u(x) \) represents the temperature of the entire cross-section of the bar at position \( x \) along the bar.
We assume that:
- The temperature of the entire cross-section at the left face is \( T_0 \).
- The temperature at the right face is \( T_L \).
- The system has reached equilibrium, meaning the heat flow and temperature distribution do not change over time.
This setup provides a classic example of a boundary](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc10cce41-ef51-4545-a724-9736e236d3b7%2F53283f81-2fc5-4179-b5af-22e65c8bed52%2F1zwxjm4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exercises
1. **Find the general solution of the following differential equations.**
a) \( x''' + x' = 0. \)
b) \( x''' + x' = 1. \)
c) \( x'''' + x'' = 0. \)
d) \( x''' - x' - 8x = 0. \)
e) \( x''' + x'' = 2e^t + 3t^2. \)
f) \( x''' - 8x = 0. \)
2. **Solve the initial value problem** \( x'' + 4x' - 4x = 0, x(0) = 1, x'(0) = 0, x''(0) = -1. \) *Hint: Guess one eigenvalue.*
3. **Write a linear, fourth-order differential equation whose general solution is:**
\[
x(t) = c_1 + c_2 t + e^{5t} (c_4 \cos 2t + c_5 \sin 5t).
\]
4. **What is the general solution of a fourth-order differential equation if the four eigenvalues are \( \lambda = 3 \pm i, 3 \pm i \)? What is the differential equation?**
---
### 2.6 Steady-State Heat Conduction*
Let us consider the following problem in steady-state heat conduction. A cylindrical, uniform, metallic bar of length \( L \) and cross-sectional area \( A \) is insulated on its lateral side. We assume the left face at \( x = 0 \) is maintained at \( T_0 \) degrees and that the right face at \( x = L \) is held at \( T_L \) degrees. What is the temperature distribution \( u = u(x) \) in the bar after it comes to equilibrium? Here \( u(x) \) represents the temperature of the entire cross-section of the bar at position \( x \) along the bar.
We assume that:
- The temperature of the entire cross-section at the left face is \( T_0 \).
- The temperature at the right face is \( T_L \).
- The system has reached equilibrium, meaning the heat flow and temperature distribution do not change over time.
This setup provides a classic example of a boundary
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