4. Consider a3(x)y"" + a2(x)y" + a₁(x)y' = 0 (a) What's one trivial solution? (b) Suppose the general solution is y(x) = C₁+C₂ cos(3x) + C3 sin(-3x). Find az(x), a₂(x), and a₁(x). (c) Suppose a3(x) = k3x³, a₂(x) = k₂x², and a₁(x) = k₁x. When does the equation have a unique solution satisfying y(0) = 1, y'(0) = 2, and y" (0) = 3?
4. Consider a3(x)y"" + a2(x)y" + a₁(x)y' = 0 (a) What's one trivial solution? (b) Suppose the general solution is y(x) = C₁+C₂ cos(3x) + C3 sin(-3x). Find az(x), a₂(x), and a₁(x). (c) Suppose a3(x) = k3x³, a₂(x) = k₂x², and a₁(x) = k₁x. When does the equation have a unique solution satisfying y(0) = 1, y'(0) = 2, and y" (0) = 3?
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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B, C only
![### Differential Equations Problem
#### Problem 4
Consider the differential equation:
\[a_3(x)y''' + a_2(x)y'' + a_1(x)y' = 0\]
##### (a) What’s one trivial solution?
Provide a solution to the differential equation that is immediately obvious or simple under given conditions.
##### (b) Suppose the general solution is \(y(x) = C_1 + C_2 \cos(3x) + C_3 \sin(-3x)\)
Find the coefficient functions \(a_3(x)\), \(a_2(x)\), and \(a_1(x)\).
##### (c) Suppose:
- \(a_3(x) = k_3 x^3\)
- \(a_2(x) = k_2 x^2\)
- \(a_1(x) = k_1 x\)
When does the equation have a unique solution satisfying the conditions:
- \(y(0) = 1\)
- \(y'(0) = 2\)
- \(y''(0) = 3\)?
**Explanation:**
1. **Trivial Solution**: When \(y(x) = 0\), the differential equation satisfies for any coefficients since there is no non-zero term to complicate it.
2. **General Solution**:
The provided general solution is \(y(x) = C_1 + C_2 \cos(3x) + C_3 \sin(-3x)\). We are tasked to identify \(a_3(x)\), \(a_2(x)\), and \(a_1(x)\) such that this general solution solves the differential equation.
3. **Specific Coefficients**:
Given the forms of \(a_3(x)\), \(a_2(x)\), and \(a_1(x)\), substitute these into the differential equation and check conditions provided for the unique solution:
- \(a_3(x) = k_3 x^3\)
- \(a_2(x) = k_2 x^2\)
- \(a_1(x) = k_1 x\)
Evaluate under initial conditions \(y(0) = 1\), \(y'(0) = 2\), and \(y''(0) = 3\) in order to find the specific values of the constants](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64590550-d676-4b57-9a24-39401c3be4ef%2Fbac054a5-2c79-4348-ba19-7c3e5bc4bab1%2Fo4gtjf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Differential Equations Problem
#### Problem 4
Consider the differential equation:
\[a_3(x)y''' + a_2(x)y'' + a_1(x)y' = 0\]
##### (a) What’s one trivial solution?
Provide a solution to the differential equation that is immediately obvious or simple under given conditions.
##### (b) Suppose the general solution is \(y(x) = C_1 + C_2 \cos(3x) + C_3 \sin(-3x)\)
Find the coefficient functions \(a_3(x)\), \(a_2(x)\), and \(a_1(x)\).
##### (c) Suppose:
- \(a_3(x) = k_3 x^3\)
- \(a_2(x) = k_2 x^2\)
- \(a_1(x) = k_1 x\)
When does the equation have a unique solution satisfying the conditions:
- \(y(0) = 1\)
- \(y'(0) = 2\)
- \(y''(0) = 3\)?
**Explanation:**
1. **Trivial Solution**: When \(y(x) = 0\), the differential equation satisfies for any coefficients since there is no non-zero term to complicate it.
2. **General Solution**:
The provided general solution is \(y(x) = C_1 + C_2 \cos(3x) + C_3 \sin(-3x)\). We are tasked to identify \(a_3(x)\), \(a_2(x)\), and \(a_1(x)\) such that this general solution solves the differential equation.
3. **Specific Coefficients**:
Given the forms of \(a_3(x)\), \(a_2(x)\), and \(a_1(x)\), substitute these into the differential equation and check conditions provided for the unique solution:
- \(a_3(x) = k_3 x^3\)
- \(a_2(x) = k_2 x^2\)
- \(a_1(x) = k_1 x\)
Evaluate under initial conditions \(y(0) = 1\), \(y'(0) = 2\), and \(y''(0) = 3\) in order to find the specific values of the constants
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