How can I verify that the functions y1 and y2 are solutions of the differential equation? And how do I know if they constitute a fundamental set of solutions?

Question

anonymous · Answer

$$y'' +4y = 0$$ 
$$y _{1} = \cos(2t)$$
$$y _{2} = \sin(2t)$$

amistre64 · Answer

how would you know of x=5 is a solution to the equation x+3 = 10 ?

anonymous · Answer

from x+3 =10, i know x =7. that means x=5 is not a solution.

amistre64 · Answer

plug them in and see of they fit ...

5+3 = 10 is not a good fit is it?

anonymous · Answer

correct.

amistre64 · Answer

saying 'i know x=7' is not mathical ... in order to prove it you have to show it

anonymous · Answer

oh, so i plug in y1 and the second derivative of y1. and then i do the same for y2.

amistre64 · Answer

yes

amistre64 · Answer

and doesnt fundamental have to do with wronskians again?

anonymous · Answer

correct. that means i will have to take the wronskian of y1 and y2 and verify that the determinant of their matrix is not equal to zero.

amistre64 · Answer

is that the only property of a fundamental solution set that we need to satisfy?  im not familiar with the properties so i wouldnt know what else to check

anonymous · Answer

Well, I am taking Diff EQ and for now, this is the only method we learned. We're only 10 lectures into class.... that is only Chapter 3 in the book. As far as my knowledge goes, that's the only way i can prove this...for now....

amistre64 · Answer

i found this ...

amistre64 · Answer

so yeah, plug and play to check that they are solutions, and Wronskian to determine if they are a fun solution set

anonymous · Answer

since the wronskin is not zero they are fundamental...thanks
by the way, where did you find that?

amistre64 · Answer

http://www.math.cuhk.edu.hk/~wei/odeas4sol.pdf

anonymous · Answer

Thanks.

amistre64 · Answer

youre welcome