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.)
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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**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.)
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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.)
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