Convert the following 2nd order IVP to a 1st order system IVP, by viewing y' as a second dependent variable called v. y" + 7y' + 5y = 9 sin (8t), y (0) = 4, y' (0) = 6.
Convert the following 2nd order IVP to a 1st order system IVP, by viewing y' as a second dependent variable called v. y" + 7y' + 5y = 9 sin (8t), y (0) = 4, y' (0) = 6.
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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Question
![**Title: Converting a Second Order Initial Value Problem (IVP) to a First Order System**
**Objective:** Learn how to convert a second order initial value problem (IVP) to a first order system by redefining the first derivative as a second dependent variable.
---
**Problem Statement:**
Convert the following 2nd order IVP to a 1st order system IVP by viewing \( y' \) as a second dependent variable called \( v \).
\[ y'' + 7y' + 5y = 9 \sin(8t) \]
\[ y(0) = 4, \quad y'(0) = 6 \]
---
**Explanation:**
1. **Introducing New Variables:**
- Let \( v = y' \). This means that \( y'' = v' \).
- This converts the second order differential equation into two first order equations:
- \( y' = v \)
- \( v' + 7v + 5y = 9 \sin(8t) \)
2. **First Order System:**
- The new system of equations becomes:
1. \( \frac{dy}{dt} = v \)
2. \( \frac{dv}{dt} = 9 \sin(8t) - 5y - 7v \)
3. **Initial Conditions:**
- Given initial conditions are:
- \( y(0) = 4 \)
- \( y'(0) = 6 \), which implies \( v(0) = 6 \)
By transforming the problem into this system, we can utilize methods and tools specific for first order systems to analyze and solve the equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd75c580-e1ed-40e9-aa84-8dd389134e22%2F115ab3e6-4d9e-4204-b06a-dd47930f09df%2F1pr59dd_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Converting a Second Order Initial Value Problem (IVP) to a First Order System**
**Objective:** Learn how to convert a second order initial value problem (IVP) to a first order system by redefining the first derivative as a second dependent variable.
---
**Problem Statement:**
Convert the following 2nd order IVP to a 1st order system IVP by viewing \( y' \) as a second dependent variable called \( v \).
\[ y'' + 7y' + 5y = 9 \sin(8t) \]
\[ y(0) = 4, \quad y'(0) = 6 \]
---
**Explanation:**
1. **Introducing New Variables:**
- Let \( v = y' \). This means that \( y'' = v' \).
- This converts the second order differential equation into two first order equations:
- \( y' = v \)
- \( v' + 7v + 5y = 9 \sin(8t) \)
2. **First Order System:**
- The new system of equations becomes:
1. \( \frac{dy}{dt} = v \)
2. \( \frac{dv}{dt} = 9 \sin(8t) - 5y - 7v \)
3. **Initial Conditions:**
- Given initial conditions are:
- \( y(0) = 4 \)
- \( y'(0) = 6 \), which implies \( v(0) = 6 \)
By transforming the problem into this system, we can utilize methods and tools specific for first order systems to analyze and solve the equation.
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