Solve: y''y' 6y = - 12t + 28 - y(0) = -6, y'(0) 21 y(t) =
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
Solve:
y''−y'−6y=−12t+28y′′-y′-6y=-12t+28
y(0)=−6, y'(0)=−21y(0)=-6, y′(0)=-21
![### Problem Statement: Solving a Second-Order Differential Equation
In this problem, we are given a second-order linear non-homogeneous differential equation along with initial conditions. The goal is to find the function \(y(t)\) that satisfies both the differential equation and the initial conditions.
#### Differential Equation
\[ y'' - y' - 6y = -12t + 28 \]
Here:
- \( y'' \) denotes the second derivative of \(y\) with respect to \(t\).
- \( y' \) denotes the first derivative of \(y\) with respect to \(t\).
- \( y \) is the unknown function of \(t\).
#### Initial Conditions
\[ y(0) = -6, \quad y'(0) = -21 \]
These conditions provide the values of \(y(t)\) and its first derivative at \(t = 0\).
#### Solution Form
\[ y(t) = \]
To solve this problem, follow these steps:
1. **Find the general solution to the homogeneous equation** associated with the given non-homogeneous equation.
2. **Determine a particular solution** to the non-homogeneous differential equation.
3. **Combine the general and particular solutions**, and use the initial conditions to find the specific constants.
4. **Verify the solution** by substituting back into the original differential equation.
After finding the solution, \(y(t)\) should be expressed in a standard form, displaying the functional relationship of \(y\) with respect to \(t\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22e84192-4b16-4dc2-817c-afb771201e2b%2F74f02f4f-66ef-45c6-b52c-118bbdd79199%2F7z6chfe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement: Solving a Second-Order Differential Equation
In this problem, we are given a second-order linear non-homogeneous differential equation along with initial conditions. The goal is to find the function \(y(t)\) that satisfies both the differential equation and the initial conditions.
#### Differential Equation
\[ y'' - y' - 6y = -12t + 28 \]
Here:
- \( y'' \) denotes the second derivative of \(y\) with respect to \(t\).
- \( y' \) denotes the first derivative of \(y\) with respect to \(t\).
- \( y \) is the unknown function of \(t\).
#### Initial Conditions
\[ y(0) = -6, \quad y'(0) = -21 \]
These conditions provide the values of \(y(t)\) and its first derivative at \(t = 0\).
#### Solution Form
\[ y(t) = \]
To solve this problem, follow these steps:
1. **Find the general solution to the homogeneous equation** associated with the given non-homogeneous equation.
2. **Determine a particular solution** to the non-homogeneous differential equation.
3. **Combine the general and particular solutions**, and use the initial conditions to find the specific constants.
4. **Verify the solution** by substituting back into the original differential equation.
After finding the solution, \(y(t)\) should be expressed in a standard form, displaying the functional relationship of \(y\) with respect to \(t\).
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