MHB Compute the Wronskian & Simplify

[shamieh]

Compute the Wronskian and simplify.
So the first part was easy

a) $y_1 = t^2 + 1$ , $y_2 = 3t^2 + k$

=$6t-2kt$

b) for what values of $k$ are the functions linearly independant

so would I just solve for $k$? I'm confused

What exactly does linearly independant mean?

 

[shamieh]

Ok so I figured out it is Linearly Independant because after I took the Wronskian I didn't get 0. But for what values of $k$ ? Would it just be as long as $k > 0$ ?

 

[Ackbach]

You can use the Wronskian to show linear independence as follows: if the Wronskian is not identically zero, and the two functions are (infinitely) differentiable, then the two functions are linearly independent. For what values of $k$ does this happen?

 

[shamieh]

You can use the Wronskian to show linear independence as follows: if the Wronskian is not identically zero, and the two functions are (infinitely) differentiable, then the two functions are linearly independent. For what values of $k$ does this happen?

as long as $k$ > 0 ?

 

[shamieh]

the two functions are inifnitely differentiable

because $y_1 = t^2 + 1 $

'= 2t + 1
'' = 2
''' = 0
''''=0
...etc

and $y_2 = 3t^2 + k$
' = 6t
'' = 6
''' = 0
etc

So would it be all values as long as $k \ne 3$

 

[shamieh]

wait so how about

t(6-2k) when k = 3 then t(6-6) = 0?

so would it be k can be any value except when k is 3?

 

[Ackbach]

wait so how about

t(6-2k) when k = 3 then t(6-6) = 0?

so would it be k can be any value except when k is 3?

You got it!