The Wronskian and the Derivative of the Wronskian

  • Thread starter sdoyle1
  • Start date
  • #1
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Homework Statement



y1(t) and y2(t), 2 solutions of the equation:

y'' +ay'+by=0, with a,b εℝ - {0}
a) Determine:
d/dt w(y1,y2)
where w(y1,y2) is the wronskian of y1(t) and y2(t)
b)
Deduce that if (y1(0),y1'(0)^T and (y2(0), y2'(0))^T are 2 linearly independent vectors. Then y1(t) and y2(t) are linearly independent functions.

Homework Equations


^T = transpose
the wronskian is the det |y1 y2|
|y1' y2'| = y1y2' -y2y1'
Vectors are linearly independent if w(y1,y2) does not equal 0

The Attempt at a Solution


For part a, do I just find the wronskian of y1 and y2 and then take the derivative?
For part b I'm super confused. I notice that if you transpose the two vectors and put them into a determinant than they are the wronskian.. other than that I'm pretty lost..
 

Answers and Replies

  • #2
179
4
for a. yes, although i think you can simplify the result a little...

for b. start with the given linear combination equal to zero...
 
  • #3
23
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I'm a bit confused regarding the linear combination. Would it just be

C1V1+C2V2=0?
I'm not sure what this would accomplish... other than c1v1=-C2V2....
 

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