# Linear Equations of Higher Order

• Apr 7th 2009, 07:05 PM
bearej50
Linear Equations of Higher Order
f(x) = 2cos(x) + 3sin(x), g(x) = 3cos(x) - 2sin(x)

Determine whether the pair of functions is linearly independent or linearly dependent on the real line.
• Apr 10th 2009, 01:35 AM
Calculus26
Use the Wronskian
See attachment
• Apr 10th 2009, 05:12 AM
bearej50
this method is invalid because f(x) and g(x) are not two solutions of any homogeneous second-order linear equation.
thanks anyway
• Apr 10th 2009, 05:35 AM
Calculus26
The Wronskian Theorem applies to any 2 functions regardless of whether or not they are solutions to a homogeneous DE.

Plain and simply if the Wronskian is not 0 2 funtions are LI.

Granted it is mainly used in regards to solutions to homogeneous equations but is not restricted to those situations.

It is easy to prove for any 2 functions if they are linearly dependent the Wronskian is 0 and hence the contrapositive I f the Wronskian is not 0 the 2 functions are LI.

with regards
• Apr 10th 2009, 05:54 AM
Calculus26
Proof
See Attacment
• Apr 10th 2009, 07:05 AM
bearej50
thank you for your help. i need to discuss this further with my professor. he lead me to believe otherwise.
• Apr 10th 2009, 07:13 AM
Calculus26
No Pblm
In all honesty I've only ever used the Wronskian in DEs.

Most prominent is it's role in solving the non-homogeneous 2d order Eq
in the method of variation of parameters.

Its easy to then associate exclusively the Wronskian to DEs but it is more general.

Having said that you may be thinking of the theorem if f1 and f2 are

solutions to a homogeneous DE and W(f1,f2) is not 0

then the general solution is Af1+Bf2.

which is really a corollary to the more general case.

take care
• Apr 12th 2009, 02:32 AM
bearej50
thanks again