4. For the DE y"-3y' x+3e", write the solution of the homogenous equation y., then write a particular solution y, in terms of coefficients A.B.C,D,... DO NOT plugin y, into the equation and do not find the coefficients A.B.C.D....
4. For the DE y"-3y' x+3e", write the solution of the homogenous equation y., then write a particular solution y, in terms of coefficients A.B.C,D,... DO NOT plugin y, into the equation and do not find the coefficients A.B.C.D....
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:**Problem Statement:**
For the differential equation (DE) \( y'' - 3y' = x + 3e^{3x} \), perform the following tasks:
1. Write the solution of the homogeneous equation \( y_c \).
2. Then write a particular solution \( y_p \) in terms of coefficients A, B, C, D, etc.
3. **Note:** Do not plug in \( y_p \) into the equation and do not find the coefficients A, B, C, D, etc.
**Explanation:**
- **Homogeneous Solution** (\( y_c \)): You need to find the complementary solution by solving the homogeneous part \( y'' - 3y' = 0 \).
- **Particular Solution** (\( y_p \)): Formulate the particular solution based on the non-homogeneous part \( x + 3e^{3x} \) using undetermined coefficients, but only express it generally with placeholders like A, B, C, D, etc., without calculating their specific values.
![**Exercise: Solving a Differential Equation**
**Problem:**
5. Find the general solution of \( y'' + 3y' + 2y = e^{5x} \).
**Hint:**
To find the solution, use the following formulas:
\[
u_1(x) = -\frac{y_2(x)f(x)}{W}
\]
\[
u_2(x) = \frac{y_1(x)f(x)}{W}
\]
Where:
- \( W \) is the Wronskian of the fundamental solutions \( y_1 \) and \( y_2 \).
- \( f(x) = e^{5x} \) in this problem.
The general solution is given by:
\[
y = u_1 y_1 + u_2 y_2
\]
**Explanation:**
Solve the homogeneous equation first to find \( y_1 \) and \( y_2 \). Then use the provided hints to find particular solutions \( u_1 \) and \( u_2 \). Finally, combine them to find the general solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5838e816-1af9-491a-a8b0-5d4d818be34b%2F21abd081-81ee-4c03-943c-e9a27fbb75ab%2F8ftjzrw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise: Solving a Differential Equation**
**Problem:**
5. Find the general solution of \( y'' + 3y' + 2y = e^{5x} \).
**Hint:**
To find the solution, use the following formulas:
\[
u_1(x) = -\frac{y_2(x)f(x)}{W}
\]
\[
u_2(x) = \frac{y_1(x)f(x)}{W}
\]
Where:
- \( W \) is the Wronskian of the fundamental solutions \( y_1 \) and \( y_2 \).
- \( f(x) = e^{5x} \) in this problem.
The general solution is given by:
\[
y = u_1 y_1 + u_2 y_2
\]
**Explanation:**
Solve the homogeneous equation first to find \( y_1 \) and \( y_2 \). Then use the provided hints to find particular solutions \( u_1 \) and \( u_2 \). Finally, combine them to find the general solution.
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