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15. y" - 4y' + 3y = 0; y(0) = 1, y'(0) = 1/3

15. y" - 4y' + 3y = 0; y(0) = 1, y'(0) = 1/3

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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15 only please.
```markdown ### Differential Equations Problems #### Problems 13–20: Solve the Given Initial Value Problem 13. \( y'' + 2y' - 8y = 0; \quad y(0) = 3, \quad y'(0) = -12 \) 14. \( y'' + y' = 0; \quad y(0) = 2, \quad y'(0) = 1 \) 15. \( y'' - 4y' + 3y = 0; \quad y(0) = 1, \quad y'(0) = \frac{1}{3} \) 16. \( y'' - 4y' - 5y = 0; \quad y(-1) = 3, \quad y'(-1) = 9 \) 17. \( y'' - 6y' + 9y = 0; \quad y(0) = 2, \quad y'(0) = 3 \) 18. \( z'' - 2z' - 2z = 0; \quad z(0) = 0, \quad z'(0) = 3 \) 19. \( y'' + 2y' + y = 0; \quad y(0) = 1, \quad y'(0) = -3 \) 20. \( y'' - 4y' + 4y = 0; \quad y(1) = 1, \quad y'(1) = 1 \) #### First-Order Constant-Coefficient Equations 21. (a) Substituting \( y = e^{rt} \), find the auxiliary equation for the first-order linear equation \( ay' + by = 0 \), where \( a \) and \( b \) are constants with \( a \neq 0 \). (b) Use the result of part (a) to find the general solution. ```
Transcribed Image Text:```markdown ### Differential Equations Problems #### Problems 13–20: Solve the Given Initial Value Problem 13. \( y'' + 2y' - 8y = 0; \quad y(0) = 3, \quad y'(0) = -12 \) 14. \( y'' + y' = 0; \quad y(0) = 2, \quad y'(0) = 1 \) 15. \( y'' - 4y' + 3y = 0; \quad y(0) = 1, \quad y'(0) = \frac{1}{3} \) 16. \( y'' - 4y' - 5y = 0; \quad y(-1) = 3, \quad y'(-1) = 9 \) 17. \( y'' - 6y' + 9y = 0; \quad y(0) = 2, \quad y'(0) = 3 \) 18. \( z'' - 2z' - 2z = 0; \quad z(0) = 0, \quad z'(0) = 3 \) 19. \( y'' + 2y' + y = 0; \quad y(0) = 1, \quad y'(0) = -3 \) 20. \( y'' - 4y' + 4y = 0; \quad y(1) = 1, \quad y'(1) = 1 \) #### First-Order Constant-Coefficient Equations 21. (a) Substituting \( y = e^{rt} \), find the auxiliary equation for the first-order linear equation \( ay' + by = 0 \), where \( a \) and \( b \) are constants with \( a \neq 0 \). (b) Use the result of part (a) to find the general solution. ```
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