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(b) Let a, b, c, xo, x1, Yo, Y1 ER and xo # Yo, a + 0. Then the ODE ay" + by' + cy satisfying y(xo) = Yo and y'(x1) = y1 is guaranteed a unique solution.

(b) Let a, b, c, xo, x1, Yo, Y1 ER and xo # Yo, a + 0. Then the ODE ay" + by' + cy satisfying y(xo) = Yo and y'(x1) = y1 is guaranteed 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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Label each of the following statements as true or false. If true, prove the statement. If false, provide a counterexample. (a) If the characteristic equation of a constant-coefficient, linear, homogeneous ODE has one non-real root, then it has at least two non-real roots. (b) Let a, b, c, xo, x1, Yo, Y1 ER and xo # Yo, a # 0. Then the ODE ay" + by' + cy satisfying y(xo) = yo and y'(x1) Yı is guaranteed a unique solution. (c) Let p and q be polynomials of degree n with domain R. If there exists cER such that W [p, q](c) = 0, then p and q are linearly dependent. (d) Suppose the general solution of a linear, second-order ODE is y = domain D, where C1, C2 E R. Then for any x e D, W[yl, y2](x) # 0. C141 + C2y2 with (e) Let L denote the Laplace transform and f (t) = sin(t) cos(t) Then L(f) exists.
Transcribed Image Text:Label each of the following statements as true or false. If true, prove the statement. If false, provide a counterexample. (a) If the characteristic equation of a constant-coefficient, linear, homogeneous ODE has one non-real root, then it has at least two non-real roots. (b) Let a, b, c, xo, x1, Yo, Y1 ER and xo # Yo, a # 0. Then the ODE ay" + by' + cy satisfying y(xo) = yo and y'(x1) Yı is guaranteed a unique solution. (c) Let p and q be polynomials of degree n with domain R. If there exists cER such that W [p, q](c) = 0, then p and q are linearly dependent. (d) Suppose the general solution of a linear, second-order ODE is y = domain D, where C1, C2 E R. Then for any x e D, W[yl, y2](x) # 0. C141 + C2y2 with (e) Let L denote the Laplace transform and f (t) = sin(t) cos(t) Then L(f) exists.
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