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annas
 3 years ago
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 ?
annas
 3 years ago
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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sami21
 3 years ago
Best ResponseYou've already chosen the best response.3it is true. let me prove this statement.

sami21
 3 years ago
Best ResponseYou've already chosen the best response.3let \[\Large f_{1}(x)=x^2\] \[\Large f_{2}(x)=x^2+5x+1\] \[\Large f_{3}=2x^2+10x+2\] 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

annas
 3 years ago
Best ResponseYou've already chosen the best response.1@sami21 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)

annas
 3 years ago
Best ResponseYou've already chosen the best response.1but i thank you @sami21 for your help :) you are really smart

sami21
 3 years ago
Best ResponseYou've already chosen the best response.3@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

annas
 3 years ago
Best ResponseYou've already chosen the best response.1no problem @sami21 but i liked that you tried to help :)
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