Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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**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)
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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)
## 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
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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
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Step 1: Take Laplace transformation on both sides of the ODE.
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Step 2: Find partial fraction decomposition.
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Step 3: Take inverse Laplace transformation .
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