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4. Find an interval around x = 0 for which the IVP (x - 2)y" + 3y = x, y(0) = 0, y'(0) = 1 has a unique solution.

4. Find an interval around x = 0 for which the IVP (x - 2)y" + 3y = x, y(0) = 0, y'(0) = 1 has a unique solution.

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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§2.1 PRELIMINARY THEORY OF HIGHER ORDER EQUATIONS PROBLEM SET 2.1 1. Given that y = ₁e²+ c₂e-² is a two-parameter family of solutions of y" - y = 0 on the interval (-∞, ∞), find a member of the family satisfying the initial conditions y(0) = 0, y′ (0) = 1. 2. Given that y = ₁x + c₂a lnx is a two-parameter family of solutions of x²y" - xy + y = 0 on the interval (0, ∞), find a member of the family satisfying the initial conditions y(1) = 3, y'(1) = -1. 3. Given that y = c₁e* cos x + c₂e³ sin z is a two-parameter family of solutions of y" - 2y + 2y = 0 on the interval (-∞, ∞), determine whether a member of the family can be found that satisfies the given conditions. (a) y(0) = 1, y'(0) = 0 (b) y(0) = 1, y(T) = -1 (c) y(0) = 1, y(π/2) = 1 (d) y(0) = 0, y(t) = 0 4. Find an interval around x = 0 for which the IVP (x - 2)y" + 3y = x, y(0) = 0, y'(0) = 1 has a unique solution. In problem 5-12, determine whether the set of functions is linearly independent on (-∞, ∞). 5. {x, x²,4x - 3x²} 6. {r,5r} 7. {sin x, cos x} 8. {x, x} 9. {sin² x, cos²x, 1} In problems 13 17, show by computing the Wronskian that the given set of functions is linearly independent on the given interval. 10. {e, e-3), (0,00) 11. {e*,re*},(-00,00) 12. {sin z, csc x}, (0, π) 13. {tan x, cotx}, (0, π/2) 14. {e,e,e¹), (-∞0,00)
Transcribed Image Text:§2.1 PRELIMINARY THEORY OF HIGHER ORDER EQUATIONS PROBLEM SET 2.1 1. Given that y = ₁e²+ c₂e-² is a two-parameter family of solutions of y" - y = 0 on the interval (-∞, ∞), find a member of the family satisfying the initial conditions y(0) = 0, y′ (0) = 1. 2. Given that y = ₁x + c₂a lnx is a two-parameter family of solutions of x²y" - xy + y = 0 on the interval (0, ∞), find a member of the family satisfying the initial conditions y(1) = 3, y'(1) = -1. 3. Given that y = c₁e* cos x + c₂e³ sin z is a two-parameter family of solutions of y" - 2y + 2y = 0 on the interval (-∞, ∞), determine whether a member of the family can be found that satisfies the given conditions. (a) y(0) = 1, y'(0) = 0 (b) y(0) = 1, y(T) = -1 (c) y(0) = 1, y(π/2) = 1 (d) y(0) = 0, y(t) = 0 4. Find an interval around x = 0 for which the IVP (x - 2)y" + 3y = x, y(0) = 0, y'(0) = 1 has a unique solution. In problem 5-12, determine whether the set of functions is linearly independent on (-∞, ∞). 5. {x, x²,4x - 3x²} 6. {r,5r} 7. {sin x, cos x} 8. {x, x} 9. {sin² x, cos²x, 1} In problems 13 17, show by computing the Wronskian that the given set of functions is linearly independent on the given interval. 10. {e, e-3), (0,00) 11. {e*,re*},(-00,00) 12. {sin z, csc x}, (0, π) 13. {tan x, cotx}, (0, π/2) 14. {e,e,e¹), (-∞0,00)
Expert Solution
Step 1

For the differential equation 

a(x) y" + b(x) y' + c(x) y + d(x) = 0

to have a unique solution, is that 

a(x) ≠ 0 .

 

 

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