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annas
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 ?
it is true. let me prove this statement.
let \[\Large f_{1}(x)=x^2\] \[\Large f_{2}(x)=x^2+5x+1\] \[\Large f_{3}=2x^2+10x+2\] here f3 is scalar multiple of 2 means \[\Large f_{3}(x)=2f_{2}(x)\] 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. http://www.wolframalpha.com/input/?i=Wronskian%5B%7Bx%5E2%2C++x%5E2+%2B+5+x+%2B+1%2C+2x%5E2+%2B+10+x+%2B+2%7D%2C+x%5D
@sami-21 no doubt on your explanation but the answer i've is: A. No. 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)
but i thank you @sami-21 for your help :) you are really smart
@annas you are right i should have assumed it y1 and y2 are non zero. if any of them is zero then one complete column will get zero.and so when complete column gets zero det gets zero :P
no problem @sami-21 but i liked that you tried to help :)