Solve the IVP y" +9y=U(t− 3) sint y (0) = 1, y (0) = 0
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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![### Solving an Initial Value Problem (IVP)
In this section, we will solve a given initial value problem (IVP).
The differential equation provided is:
\[ y'' + 9y = U(t - 3\pi) \sin t \]
with the initial conditions:
\[ y(0) = 1 \]
\[ y'(0) = 0 \]
where \( U(t - 3\pi) \) represents the unit step function centered at \( t = 3\pi \).
#### Explanation:
1. **Differential Equation**: The given equation \( y'' + 9y = U(t - 3\pi) \sin t \) is a second-order linear differential equation with a non-homogeneous term \( U(t - 3\pi) \sin t \).
2. **Unit Step Function**: \( U(t - 3\pi) \) is the Heaviside Step Function, which means the function \( \sin t \) will be "turned on" at \( t = 3\pi \).
3. **Initial Conditions**: These conditions specify the value of the function \( y(t) \) and its first derivative \( y'(t) \) at \( t = 0 \). Specifically, \( y(0) = 1 \) and \( y'(0) = 0 \).
The solution approach would typically include solving the corresponding homogeneous equation and finding a particular solution for the non-homogeneous part. Additionally, the initial conditions will help determine the constants involved.
Understanding and solving differential equations with initial values is a crucial skill in mathematical modeling and analysis, helping describe a wide range of physical and theoretical systems accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3acf6a6-30f5-4bc7-ac16-270edca78ecc%2F1ba32350-5195-4f5c-8343-e3dfd88e43eb%2Fitlyblj_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving an Initial Value Problem (IVP)
In this section, we will solve a given initial value problem (IVP).
The differential equation provided is:
\[ y'' + 9y = U(t - 3\pi) \sin t \]
with the initial conditions:
\[ y(0) = 1 \]
\[ y'(0) = 0 \]
where \( U(t - 3\pi) \) represents the unit step function centered at \( t = 3\pi \).
#### Explanation:
1. **Differential Equation**: The given equation \( y'' + 9y = U(t - 3\pi) \sin t \) is a second-order linear differential equation with a non-homogeneous term \( U(t - 3\pi) \sin t \).
2. **Unit Step Function**: \( U(t - 3\pi) \) is the Heaviside Step Function, which means the function \( \sin t \) will be "turned on" at \( t = 3\pi \).
3. **Initial Conditions**: These conditions specify the value of the function \( y(t) \) and its first derivative \( y'(t) \) at \( t = 0 \). Specifically, \( y(0) = 1 \) and \( y'(0) = 0 \).
The solution approach would typically include solving the corresponding homogeneous equation and finding a particular solution for the non-homogeneous part. Additionally, the initial conditions will help determine the constants involved.
Understanding and solving differential equations with initial values is a crucial skill in mathematical modeling and analysis, helping describe a wide range of physical and theoretical systems accurately.
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