A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. y” - y=12t, yp(t) = - 12t The general solution is y(t) = (Do not use d, D, e, E, i, or I as arbitrary constants since these letters already have defined meanings.)
A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. y” - y=12t, yp(t) = - 12t The general solution is y(t) = (Do not use d, D, e, E, i, or I as arbitrary constants since these letters already have defined meanings.)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 12E
Related questions
Question
3Differential equations
Please help solve show work
Thank You.
![---
**Finding the General Solution of a Nonhomogeneous Equation**
### Problem Statement
Given a nonhomogeneous equation and a particular solution, the goal is to find the general solution for the equation:
\[ y'' - y = 12t, \quad y_p(t) = -12t \]
### Solution Guidelines
The general solution for \( y(t) \) comprises two parts: the general solution of the corresponding homogeneous equation and the particular solution to the nonhomogeneous equation.
#### 1. Solve the Homogeneous Equation
First, solve the corresponding homogeneous equation:
\[ y'' - y = 0 \]
The characteristic equation for this differential equation is:
\[ r^2 - 1 = 0 \]
This factors as:
\[ (r - 1)(r + 1) = 0 \]
Thus, the roots are \( r = 1 \) and \( r = -1 \).
Therefore, the general solution to the homogeneous equation is:
\[ y_h(t) = C_1 e^t + C_2 e^{-t} \]
#### 2. Incorporate the Particular Solution
Given the particular solution to the nonhomogeneous equation:
\[ y_p(t) = -12t \]
#### 3. Form the General Solution
Combine the solutions obtained:
\[ y(t) = y_h(t) + y_p(t) \]
\[ y(t) = C_1 e^t + C_2 e^{-t} - 12t \]
In conclusion, the general solution to the given nonhomogeneous differential equation is:
\[ y(t) = C_1 e^t + C_2 e^{-t} - 12t \]
(Note: It is specified not to use \( d \), \( D \), \( e \), \( E \), \( i \), or \( I \) as arbitrary constants since these letters already have defined meanings.)
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1d700aa-072a-4ff7-be1d-cdca0f10aa9d%2F1f8834ee-e02a-44e7-b881-db1ff4bb10da%2F22wlcdr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Finding the General Solution of a Nonhomogeneous Equation**
### Problem Statement
Given a nonhomogeneous equation and a particular solution, the goal is to find the general solution for the equation:
\[ y'' - y = 12t, \quad y_p(t) = -12t \]
### Solution Guidelines
The general solution for \( y(t) \) comprises two parts: the general solution of the corresponding homogeneous equation and the particular solution to the nonhomogeneous equation.
#### 1. Solve the Homogeneous Equation
First, solve the corresponding homogeneous equation:
\[ y'' - y = 0 \]
The characteristic equation for this differential equation is:
\[ r^2 - 1 = 0 \]
This factors as:
\[ (r - 1)(r + 1) = 0 \]
Thus, the roots are \( r = 1 \) and \( r = -1 \).
Therefore, the general solution to the homogeneous equation is:
\[ y_h(t) = C_1 e^t + C_2 e^{-t} \]
#### 2. Incorporate the Particular Solution
Given the particular solution to the nonhomogeneous equation:
\[ y_p(t) = -12t \]
#### 3. Form the General Solution
Combine the solutions obtained:
\[ y(t) = y_h(t) + y_p(t) \]
\[ y(t) = C_1 e^t + C_2 e^{-t} - 12t \]
In conclusion, the general solution to the given nonhomogeneous differential equation is:
\[ y(t) = C_1 e^t + C_2 e^{-t} - 12t \]
(Note: It is specified not to use \( d \), \( D \), \( e \), \( E \), \( i \), or \( I \) as arbitrary constants since these letters already have defined meanings.)
---
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage