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 59E: According to the solution in Exercise 58 of the differential equation for Newtons law of cooling,...
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![### Solving Differential Equations by Variation of Parameters
To solve the differential equation using the method of variation of parameters, follow the steps outlined below.
Consider the given differential equation:
\[ 2y'' - 4y' + 4y = e^x \sec x \]
Here, \(y''\) denotes the second derivative of \(y\) with respect to \(x\), and \(y'\) denotes the first derivative of \(y\) with respect to \(x\).
### Steps to Solve by Variation of Parameters:
1. **Find the Complementary Solution (Homogeneous Solution):**
First, solve the homogeneous part of the differential equation:
\[ 2y'' - 4y' + 4y = 0 \]
2. **Determine the Particular Solution:**
Next, use the variation of parameters method to find the particular solution to the non-homogeneous equation:
\[ 2y'' - 4y' + 4y = e^x \sec x \]
3. **Construct the General Solution:**
Finally, combine the complementary solution and the particular solution to form the general solution.
### Provide Your Solution:
\[ y(x) = \boxed{} \]
**Note:** Ensure to complete the steps of the variation of parameters method to obtain the solution for \(y(x)\). Fill in the box with the final expression you derive.
### Diagram Explanation:
There are no diagrams or graphs associated with this content. The primary focus here is on solving the differential equation mathematically using the variation of parameters method.
For a thorough explanation on variation of parameters, including illustrative examples, please refer to our differential equations section.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c333d55-482b-4cad-8212-73eff6844637%2Ff1e0af46-7612-4bcd-af47-ed9ff43d2168%2Fjlpv079_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Differential Equations by Variation of Parameters
To solve the differential equation using the method of variation of parameters, follow the steps outlined below.
Consider the given differential equation:
\[ 2y'' - 4y' + 4y = e^x \sec x \]
Here, \(y''\) denotes the second derivative of \(y\) with respect to \(x\), and \(y'\) denotes the first derivative of \(y\) with respect to \(x\).
### Steps to Solve by Variation of Parameters:
1. **Find the Complementary Solution (Homogeneous Solution):**
First, solve the homogeneous part of the differential equation:
\[ 2y'' - 4y' + 4y = 0 \]
2. **Determine the Particular Solution:**
Next, use the variation of parameters method to find the particular solution to the non-homogeneous equation:
\[ 2y'' - 4y' + 4y = e^x \sec x \]
3. **Construct the General Solution:**
Finally, combine the complementary solution and the particular solution to form the general solution.
### Provide Your Solution:
\[ y(x) = \boxed{} \]
**Note:** Ensure to complete the steps of the variation of parameters method to obtain the solution for \(y(x)\). Fill in the box with the final expression you derive.
### Diagram Explanation:
There are no diagrams or graphs associated with this content. The primary focus here is on solving the differential equation mathematically using the variation of parameters method.
For a thorough explanation on variation of parameters, including illustrative examples, please refer to our differential equations section.
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