Solve y" + y'-6y=2x
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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![## Solving the Differential Equation: \( y'' + y' - 6y = 2x \)
### Step-by-step Solution:
1. **Solve the homogeneous equation**:
\[ y'' + y' - 6y = 0 \]
Let's denote the complementary function as \( y_c \).
Using (i):
\[ y_c = y_{1} + y_{2} \]
\[ y_c = C_{1}y_{1} + C_{2}y_{2} \]
2. **Find the particular solution**:
Using (ii):
\[ y = Ax + B \ \text{(could be)} \]
_Evaluate if duplication, is it 1st Order or 2nd Order?_
The particular solution \( y_p = (Ax + B)x \), and then proceed with the general solution.
### Detailed Explanation:
- **Graphs or Diagrams:**
While no graphs or diagrams are present in the provided image, it is important when solving differential equations, especially when dealing with their graphical solutions, to:
1. Plot the complementary functions \( y_1 \) and \( y_2 \).
2. Plot the particular solution \( y_p \).
3. Combine these to visualize the general solution and confirm the behavior of the differential equation.
In educational contexts, ensure to:
- Introduce the steps clearly when breaking down the problem.
- Encourage students to verify solutions by substituting back into the original equation.
- Ensure students are familiar with differentiation and integration techniques necessary to solve these equations.
### Note:
The process involves verifying if the trial solution \( y = Ax + B \) works by substituting it into the non-homogeneous equation. If it results in redundancy or simplification issues, adjust the form by multiplying by \( x \).
By following these guidelines, you can successfully approach and solve differential equations step by step.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c88e3a7-d6d9-4180-9e78-6a7df1a2f887%2Faa673dbe-c309-4d67-aaa3-d8a586ee3f44%2Ffhi15vo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Solving the Differential Equation: \( y'' + y' - 6y = 2x \)
### Step-by-step Solution:
1. **Solve the homogeneous equation**:
\[ y'' + y' - 6y = 0 \]
Let's denote the complementary function as \( y_c \).
Using (i):
\[ y_c = y_{1} + y_{2} \]
\[ y_c = C_{1}y_{1} + C_{2}y_{2} \]
2. **Find the particular solution**:
Using (ii):
\[ y = Ax + B \ \text{(could be)} \]
_Evaluate if duplication, is it 1st Order or 2nd Order?_
The particular solution \( y_p = (Ax + B)x \), and then proceed with the general solution.
### Detailed Explanation:
- **Graphs or Diagrams:**
While no graphs or diagrams are present in the provided image, it is important when solving differential equations, especially when dealing with their graphical solutions, to:
1. Plot the complementary functions \( y_1 \) and \( y_2 \).
2. Plot the particular solution \( y_p \).
3. Combine these to visualize the general solution and confirm the behavior of the differential equation.
In educational contexts, ensure to:
- Introduce the steps clearly when breaking down the problem.
- Encourage students to verify solutions by substituting back into the original equation.
- Ensure students are familiar with differentiation and integration techniques necessary to solve these equations.
### Note:
The process involves verifying if the trial solution \( y = Ax + B \) works by substituting it into the non-homogeneous equation. If it results in redundancy or simplification issues, adjust the form by multiplying by \( x \).
By following these guidelines, you can successfully approach and solve differential equations step by step.
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