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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
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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### 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
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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