Q. Is it true that the Wronskian of two functions y1 and y2 is zero if and only if one of these two functions is a scalor multiple of the other. Explain with proof ?
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it is true. let me prove this statement.
here f3 is scalar multiple of 2 means
you can calculate its Wronskian . it should give you zero Because they are linearly dependent . and for Linearly dependent function Wronskian is always zero.
here you can verify this on Wolfram.
@sami-21 no doubt on your explanation but the answer i've is:
We have seen that if y1 or y2 is never zero, then your statement is true. Indeed, suppose y2 is never zero. Then the quotient y1 / y2
is constant if its derivative -W/(y2)2 is zero. But the derivative -W/(y2)2 is zero if and only if W=0.
However, without assuming that y1 or y2 is never zero, your statement is not true.
Example: Let y1(t)=t2. Let y2(t) be the function which is -t2 on the interval (-∞, 0) and which is t2 on the interval [0, ∞). Then the Wronskian is everywhere zero, but the functions y1 and y2 are not proportional.
from the source (http://www.math.dartmouth.edu/archive/m23s09/public_html/sadykov/qa.html)