Use Laplace transforms to solve the following initial value problem. x'' + 4x' + 13x = te¯¹; x(0) = 0, x'(0) = 4
Use Laplace transforms to solve the following initial value problem. x'' + 4x' + 13x = te¯¹; x(0) = 0, x'(0) = 4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
![**Problem Statement:**
Use Laplace transforms to solve the following initial value problem.
\[ x'' + 4x' + 13x = te^{-t} \]
Initial conditions:
- \( x(0) = 0 \)
- \( x'(0) = 4 \)
**Solution:**
\[ x(t) = \ldots \] (Solution to be determined using Laplace transforms)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3200e892-7b83-4670-8aa5-c4c84f2a6adb%2F2e19edb2-4be7-488b-be43-06333d2e864c%2Fviyl72o_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use Laplace transforms to solve the following initial value problem.
\[ x'' + 4x' + 13x = te^{-t} \]
Initial conditions:
- \( x(0) = 0 \)
- \( x'(0) = 4 \)
**Solution:**
\[ x(t) = \ldots \] (Solution to be determined using Laplace transforms)

Transcribed Image Text:## Laplace Transform Pairs
The table below provides a list of common Laplace transforms and their inverse transforms, useful for various applications in engineering and physics.
### Laplace Transform Formulas
#### Left Side: Basic Functions
1. **Constant Function (1):**
- \( f(t) = 1 \Rightarrow \mathcal{L}\{f(t)\} = \frac{1}{s} \quad (s > 0) \)
2. **Linear Function (t):**
- \( f(t) = t \Rightarrow \mathcal{L}\{f(t)\} = \frac{1}{s^2} \quad (s > 0) \)
3. **Power Function (\(t^n\)):**
- \( f(t) = t^n \Rightarrow \mathcal{L}\{f(t)\} = \frac{n!}{s^{n+1}} \quad (s > 0) \)
4. **General Power Function (\(t^a\)):**
- \( f(t) = t^a \Rightarrow \mathcal{L}\{f(t)\} = \frac{\Gamma(a+1)}{s^{n+1}} \quad (s > 0) \)
5. **Exponential Function (\(e^{at}\)):**
- \( f(t) = e^{at} \Rightarrow \mathcal{L}\{f(t)\} = \frac{1}{s-a} \quad (s > a) \)
#### Right Side: Trigonometric and Hyperbolic Functions
1. **Cosine Function (\(\cos kt\)):**
- \( f(t) = \cos kt \Rightarrow \mathcal{L}\{f(t)\} = \frac{s}{s^2 + k^2} \quad (s > 0) \)
2. **Sine Function (\(\sin kt\)):**
- \( f(t) = \sin kt \Rightarrow \mathcal{L}\{f(t)\} = \frac{k}{s^2 + k^2} \quad (s > 0) \)
3. **Hyperbolic Cosine Function (\(\cosh kt\)):**
- \( f(t) = \cosh kt \Rightarrow \mathcal{L}\{f
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 5 images

Recommended textbooks for you

Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated

Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY

Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated

Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY

Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,

