Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 17E: Find the general solution for each differential equation. Verify that each solution satisfies the...
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![### Solving Differential Equations by Undetermined Coefficients
**Problem Statement:**
Solve the given differential equation by the method of undetermined coefficients.
\[ y'' - y' = -6 \]
**Solution Process:**
_y(x) = ______________
In this problem, you are tasked with solving the differential equation \( y'' - y' = -6 \) using the method of undetermined coefficients. The method of undetermined coefficients is a technique to find particular solutions to non-homogeneous linear differential equations with constant coefficients.
**Explanation of the Method:**
1. **Find the complementary solution** \( y_c(x) \):
- Solve the associated homogeneous equation \( y'' - y' = 0 \).
- The characteristic equation for this differential equation is obtained by replacing \( y'' \) with \( r^2 \) and \( y' \) with \( r \), resulting in \( r^2 - r = 0 \). Solving this quadratic equation gives the roots \( r = 0 \) and \( r = 1 \).
- Therefore, the general solution to the homogeneous equation is:
\[
y_c(x) = C_1 e^{0x} + C_2 e^{1x} \implies y_c(x) = C_1 + C_2 e^x
\]
2. **Find the particular solution** \( y_p(x) \):
- For the non-homogeneous term \(-6\), guess a particular solution. Since the term is a constant, try \( y_p(x) = A \). Substitute \( y_p \) into the differential equation to determine \( A \).
- Substituting \( y_p = A \) into \( y'' - y' = -6 \):
\[
0 - 0 = -6
\]
This suggests choosing \( y_p = A \) where \( A = -6 \).
3. **Combine the solutions**:
- The general solution to the differential equation is the sum of the complementary and particular solutions:
\[
y(x) = y_c(x) + y_p(x) = C_1 + C_2 e^x - 6
\]
Fill in the blank box with the final expression to assist with the problem-solving process.
**Answer:**
\[ y(x) = C_1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c333d55-482b-4cad-8212-73eff6844637%2Fdf4e0e44-088e-4458-9aca-ce7b593d1d95%2Fhc0cuud_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Differential Equations by Undetermined Coefficients
**Problem Statement:**
Solve the given differential equation by the method of undetermined coefficients.
\[ y'' - y' = -6 \]
**Solution Process:**
_y(x) = ______________
In this problem, you are tasked with solving the differential equation \( y'' - y' = -6 \) using the method of undetermined coefficients. The method of undetermined coefficients is a technique to find particular solutions to non-homogeneous linear differential equations with constant coefficients.
**Explanation of the Method:**
1. **Find the complementary solution** \( y_c(x) \):
- Solve the associated homogeneous equation \( y'' - y' = 0 \).
- The characteristic equation for this differential equation is obtained by replacing \( y'' \) with \( r^2 \) and \( y' \) with \( r \), resulting in \( r^2 - r = 0 \). Solving this quadratic equation gives the roots \( r = 0 \) and \( r = 1 \).
- Therefore, the general solution to the homogeneous equation is:
\[
y_c(x) = C_1 e^{0x} + C_2 e^{1x} \implies y_c(x) = C_1 + C_2 e^x
\]
2. **Find the particular solution** \( y_p(x) \):
- For the non-homogeneous term \(-6\), guess a particular solution. Since the term is a constant, try \( y_p(x) = A \). Substitute \( y_p \) into the differential equation to determine \( A \).
- Substituting \( y_p = A \) into \( y'' - y' = -6 \):
\[
0 - 0 = -6
\]
This suggests choosing \( y_p = A \) where \( A = -6 \).
3. **Combine the solutions**:
- The general solution to the differential equation is the sum of the complementary and particular solutions:
\[
y(x) = y_c(x) + y_p(x) = C_1 + C_2 e^x - 6
\]
Fill in the blank box with the final expression to assist with the problem-solving process.
**Answer:**
\[ y(x) = C_1
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