Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Title: Solving Non-Homogeneous Ordinary Differential Equations Using the Method of Undetermined Coefficients**
**Introduction:**
In this exercise, we will learn how to find a particular solution to a non-homogeneous ordinary differential equation (ODE) using the method of undetermined coefficients.
**Problem Statement:**
Find a particular solution of the non-homogeneous ODE using the undetermined coefficients method:
\[ y'' + 4y = (5x^2 - x + 10)e^x. \]
**Explanation:**
In the given problem, we have a second-order linear differential equation with constant coefficients on the left-hand side, and a non-homogeneous term \((5x^2 - x + 10)e^x\) on the right-hand side.
**Method Overview:**
The method of undetermined coefficients involves choosing a suitable form for the particular solution based on the non-homogeneous term. This form includes constants (undetermined coefficients) which will be solved by substituting back into the differential equation.
**Steps to Solve:**
1. Identify the form of the non-homogeneous term, \((5x^2 - x + 10)e^x\), which suggests that the particular solution \(y_p\) should also be of the form \( (Ax^2 + Bx + C)e^x\).
2. Substitute the assumed particular solution into the differential equation.
3. Calculate the derivatives needed: \(y_p'\) and \(y_p''\).
4. Substitute back into the left-hand side of the differential equation, and equate coefficients with the right-hand side to solve for \(A\), \(B\), and \(C\).
5. Combine the particular solution \(y_p\) with the general solution of the corresponding homogeneous equation to form the complete solution.
**Graphical Representation:**
If graphs or diagrams are required, you can plot the function \( (5x^2 - x + 10)e^x \) as well as the resulting particular solution \(y_p\) to visually verify the correct form and solution behavior.
**Conclusion:**
The method of undetermined coefficients is a powerful tool for solving non-homogeneous linear differential equations, especially when the non-homogeneous part is a combination of polynomial, exponential, sine, or cosine functions.](https://content.bartleby.com/qna-images/question/47094caa-49f2-44b6-998c-cb271f00b44a/b7d73481-e894-4ad1-b710-af186e5d6e3f/01hhh8_processed.png)
Transcribed Image Text:**Title: Solving Non-Homogeneous Ordinary Differential Equations Using the Method of Undetermined Coefficients**
**Introduction:**
In this exercise, we will learn how to find a particular solution to a non-homogeneous ordinary differential equation (ODE) using the method of undetermined coefficients.
**Problem Statement:**
Find a particular solution of the non-homogeneous ODE using the undetermined coefficients method:
\[ y'' + 4y = (5x^2 - x + 10)e^x. \]
**Explanation:**
In the given problem, we have a second-order linear differential equation with constant coefficients on the left-hand side, and a non-homogeneous term \((5x^2 - x + 10)e^x\) on the right-hand side.
**Method Overview:**
The method of undetermined coefficients involves choosing a suitable form for the particular solution based on the non-homogeneous term. This form includes constants (undetermined coefficients) which will be solved by substituting back into the differential equation.
**Steps to Solve:**
1. Identify the form of the non-homogeneous term, \((5x^2 - x + 10)e^x\), which suggests that the particular solution \(y_p\) should also be of the form \( (Ax^2 + Bx + C)e^x\).
2. Substitute the assumed particular solution into the differential equation.
3. Calculate the derivatives needed: \(y_p'\) and \(y_p''\).
4. Substitute back into the left-hand side of the differential equation, and equate coefficients with the right-hand side to solve for \(A\), \(B\), and \(C\).
5. Combine the particular solution \(y_p\) with the general solution of the corresponding homogeneous equation to form the complete solution.
**Graphical Representation:**
If graphs or diagrams are required, you can plot the function \( (5x^2 - x + 10)e^x \) as well as the resulting particular solution \(y_p\) to visually verify the correct form and solution behavior.
**Conclusion:**
The method of undetermined coefficients is a powerful tool for solving non-homogeneous linear differential equations, especially when the non-homogeneous part is a combination of polynomial, exponential, sine, or cosine functions.
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