Solve the given differential equation by undetermined coefficients. y" 2y' = 2x + 3 – 2-2x y(x) = | %3D
Solve the given differential equation by undetermined coefficients. y" 2y' = 2x + 3 – 2-2x y(x) = | %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Title: Solving Differential Equations Using Undetermined Coefficients**
**Problem:**
Solve the given differential equation by undetermined coefficients.
\[ y'' + 2y' = 2x + 3 - e^{-2x} \]
**Solution:**
\[ y(x) = \_ \]
---
**Explanation:**
To solve this differential equation, we will use the method of undetermined coefficients. This technique involves finding a particular solution to the non-homogeneous equation by assuming a form similar to the non-homogeneous part on the right-hand side.
1. **Identify the Homogeneous Equation:**
- Start by solving the corresponding homogeneous equation:
\[ y'' + 2y' = 0 \]
- Find the complementary solution, \( y_c(x) \).
2. **Particular Solution Method:**
- Assume a form for the particular solution, \( y_p(x) \), based on the right-hand side of the original equation, which consists of a polynomial part and an exponential part.
- For the polynomial part \( 2x + 3 \), assume \( y_p(x) = Ax + B \).
- For the exponential part \( -e^{-2x} \), assume \( y_p(x) = Ce^{-2x} \).
3. **Solve for Coefficients:**
- Substitute \( y_p(x) = Ax + B + Ce^{-2x} \) into the original differential equation and equate coefficients to solve for \( A \), \( B \), and \( C \).
4. **General Solution:**
- Combine the complementary and particular solutions to determine the general solution of the differential equation:
\[ y(x) = y_c(x) + y_p(x) \]
This method efficiently provides solutions for linear differential equations with easily assumed forms for their non-homogeneous parts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F711908d8-7456-4dad-baa5-347305ed075d%2Fe60b2ead-ed05-412a-9eb5-ff827ef5544d%2Fwav78til_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Differential Equations Using Undetermined Coefficients**
**Problem:**
Solve the given differential equation by undetermined coefficients.
\[ y'' + 2y' = 2x + 3 - e^{-2x} \]
**Solution:**
\[ y(x) = \_ \]
---
**Explanation:**
To solve this differential equation, we will use the method of undetermined coefficients. This technique involves finding a particular solution to the non-homogeneous equation by assuming a form similar to the non-homogeneous part on the right-hand side.
1. **Identify the Homogeneous Equation:**
- Start by solving the corresponding homogeneous equation:
\[ y'' + 2y' = 0 \]
- Find the complementary solution, \( y_c(x) \).
2. **Particular Solution Method:**
- Assume a form for the particular solution, \( y_p(x) \), based on the right-hand side of the original equation, which consists of a polynomial part and an exponential part.
- For the polynomial part \( 2x + 3 \), assume \( y_p(x) = Ax + B \).
- For the exponential part \( -e^{-2x} \), assume \( y_p(x) = Ce^{-2x} \).
3. **Solve for Coefficients:**
- Substitute \( y_p(x) = Ax + B + Ce^{-2x} \) into the original differential equation and equate coefficients to solve for \( A \), \( B \), and \( C \).
4. **General Solution:**
- Combine the complementary and particular solutions to determine the general solution of the differential equation:
\[ y(x) = y_c(x) + y_p(x) \]
This method efficiently provides solutions for linear differential equations with easily assumed forms for their non-homogeneous parts.
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