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
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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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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