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True or False, and provide a brief explanation. (a) If y1(t) and y2(t) are both twice continuously differentiable on the real line and their Wronskian is W(y1,Y2) = 3t²e*, then there are no functions p(t) and q(t), both continuous in the interval (-1,1) such that y, and y2 are both solutions of the equation y"+p(t)y'+q(t)y = 0. (b) If b<0, then lim»» y(t) = ∞ or lim» y(t) = - ∞ for every non-constant solution, y(t), of the equation y"tay'+by = 0. (c) There is a d>0, such that the initial value problem y'=In(t+y+2), y(0) = 0 has a unique solution defined in the interval (-8,8).

True or False, and provide a brief explanation. (a) If y1(t) and y2(t) are both twice continuously differentiable on the real line and their Wronskian is W(y1,Y2) = 3t²e*, then there are no functions p(t) and q(t), both continuous in the interval (-1,1) such that y, and y2 are both solutions of the equation y"+p(t)y'+q(t)y = 0. (b) If b<0, then lim»» y(t) = ∞ or lim» y(t) = - ∞ for every non-constant solution, y(t), of the equation y"tay'+by = 0. (c) There is a d>0, such that the initial value problem y'=In(t+y+2), y(0) = 0 has a unique solution defined in the interval (-8,8).

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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True or False, and provide a brief explanation. (a) If y1(t) and y2(t) are both twice continuously differentiable on the real line and their Wronskian is W(y1,Y2) = 3t²e*, then there are no functions p(t) and q(t), both continuous in the interval (-1,1) such that y, and y2 are both solutions of the equation y"+p(t)y'+q(t)y = 0. (b) If b<0, then lim»» y(t) = ∞ or lim» y(t) = - ∞ for every non-constant solution, y(t), of the equation y"tay'+by = 0. (c) There is a d>0, such that the initial value problem y'=In(t+y+2), y(0) = 0 has a unique solution defined in the interval (-8,8).
Transcribed Image Text:True or False, and provide a brief explanation. (a) If y1(t) and y2(t) are both twice continuously differentiable on the real line and their Wronskian is W(y1,Y2) = 3t²e*, then there are no functions p(t) and q(t), both continuous in the interval (-1,1) such that y, and y2 are both solutions of the equation y"+p(t)y'+q(t)y = 0. (b) If b<0, then lim»» y(t) = ∞ or lim» y(t) = - ∞ for every non-constant solution, y(t), of the equation y"tay'+by = 0. (c) There is a d>0, such that the initial value problem y'=In(t+y+2), y(0) = 0 has a unique solution defined in the interval (-8,8).
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