2. (Section 5.6, problem 11) Find the solution of the following IVP: y" + 2y' + 10y h(t – 2), y(0) y'(0) 0, 0.
2. (Section 5.6, problem 11) Find the solution of the following IVP: y" + 2y' + 10y h(t – 2), y(0) y'(0) 0, 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Section 5.6, Problem 11: Solving the Initial Value Problem (IVP)
**Problem Statement:**
Find the solution of the following IVP:
\[
y'' + 2y' + 10y = h(t - 2),
\]
with the initial conditions:
\[
y(0) = 0, \quad y'(0) = 0.
\]
**Solution Approach:**
The problem mentions that the Laplace transform of the solution is given by:
\[
Y(s) = \frac{e^{-2s}}{s(s^2 + 2s + 10)}.
\]
**Additional Information:**
Note that the source reference uses \(u(t)\) in place of \(h(t)\).
### Explanation:
- **Equation:** The given differential equation is a second-order linear ordinary differential equation with constant coefficients.
- **Initial Conditions:** The conditions \(y(0) = 0\) and \(y'(0) = 0\) imply that initially, both the function and its first derivative are zero.
- **Laplace Transform:** The Laplace transform is used to convert differential equations into algebraic equations in the complex frequency domain, which are often easier to solve.
- **Expression for \(Y(s)\):** The solution in the s-domain shows the impact of a delayed impulse, represented by the exponential term \(e^{-2s}\), and a denominator indicative of a second-order system.
### Key Concepts:
- **Laplace Transform in Control Systems:** This approach is widely used in control systems to handle initial conditions and delay terms efficiently.
- **Interpretation of Poles:** The denominator \(s(s^2 + 2s + 10)\) suggests a system with one real pole at \(s=0\) and two complex poles, which determine the system's transient response.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40e201e1-8e56-40e1-84d9-d5681b3b131e%2Fb72d8e05-3711-44e5-bd21-66d2b4d2a41a%2F2hz99fn_processed.png&w=3840&q=75)
Transcribed Image Text:### Section 5.6, Problem 11: Solving the Initial Value Problem (IVP)
**Problem Statement:**
Find the solution of the following IVP:
\[
y'' + 2y' + 10y = h(t - 2),
\]
with the initial conditions:
\[
y(0) = 0, \quad y'(0) = 0.
\]
**Solution Approach:**
The problem mentions that the Laplace transform of the solution is given by:
\[
Y(s) = \frac{e^{-2s}}{s(s^2 + 2s + 10)}.
\]
**Additional Information:**
Note that the source reference uses \(u(t)\) in place of \(h(t)\).
### Explanation:
- **Equation:** The given differential equation is a second-order linear ordinary differential equation with constant coefficients.
- **Initial Conditions:** The conditions \(y(0) = 0\) and \(y'(0) = 0\) imply that initially, both the function and its first derivative are zero.
- **Laplace Transform:** The Laplace transform is used to convert differential equations into algebraic equations in the complex frequency domain, which are often easier to solve.
- **Expression for \(Y(s)\):** The solution in the s-domain shows the impact of a delayed impulse, represented by the exponential term \(e^{-2s}\), and a denominator indicative of a second-order system.
### Key Concepts:
- **Laplace Transform in Control Systems:** This approach is widely used in control systems to handle initial conditions and delay terms efficiently.
- **Interpretation of Poles:** The denominator \(s(s^2 + 2s + 10)\) suggests a system with one real pole at \(s=0\) and two complex poles, which determine the system's transient response.
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