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If x(t) and y(t) are solutions to the same homogeneous linear differential equation, it can be shown that their Wronskian W(t)=Ce^A(t) for some function A(t) and constant C. Thus W is either zero everywhere or zero nowhere. If it is zero everywhere then x(t) and y(t) are linearly dependent. However if x(t) and y(t) are not solutions to the same homogeneous linear differential equation this is not necessarily the case. Calculate the Wronskian of x^2, xlx on the interval (-1,1). Then check directly (without using their Wronskian) if they are linearly independent or not. Now, True or False? x^2, xlx are linearly independent on (-1,1) (If you get this correct you've done better than this mathematician.) True False

If x(t) and y(t) are solutions to the same homogeneous linear differential equation, it can be shown that their Wronskian W(t)=Ce^A(t) for some function A(t) and constant C. Thus W is either zero everywhere or zero nowhere. If it is zero everywhere then x(t) and y(t) are linearly dependent. However if x(t) and y(t) are not solutions to the same homogeneous linear differential equation this is not necessarily the case. Calculate the Wronskian of x^2, xlx on the interval (-1,1). Then check directly (without using their Wronskian) if they are linearly independent or not. Now, True or False? x^2, xlx are linearly independent on (-1,1) (If you get this correct you've done better than this mathematician.) True False

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 12CR
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Question 4 If x(t) and y(t) are solutions to the same homogeneous linear differential equation, it can be shown that their Wronskian W(t)=Ce^A(t) for some function A(t) and constant C. Thus W is either zero everywhere or zero nowhere. If it is zero everywhere then x(t) and y(t) are linearly dependent. However if x(t) and y(t) are not solutions to the same homogeneous linear differential equation this is not necessarily the case. Calculate the Wronskian of x^2, xlx on the interval (-1,1). Then check directly (without using their Wronskian) if they are linearly independent or not. Now, True or False? x^2, xlx are linearly independent on (-1,1) (If you get this correct you've done better than this mathematician.) True False
Transcribed Image Text:Question 4 If x(t) and y(t) are solutions to the same homogeneous linear differential equation, it can be shown that their Wronskian W(t)=Ce^A(t) for some function A(t) and constant C. Thus W is either zero everywhere or zero nowhere. If it is zero everywhere then x(t) and y(t) are linearly dependent. However if x(t) and y(t) are not solutions to the same homogeneous linear differential equation this is not necessarily the case. Calculate the Wronskian of x^2, xlx on the interval (-1,1). Then check directly (without using their Wronskian) if they are linearly independent or not. Now, True or False? x^2, xlx are linearly independent on (-1,1) (If you get this correct you've done better than this mathematician.) True False
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