Finding a second order ODE from complex exponential solutions

[bobey]

show that y1(x) = e^(2+i)x and y2(x) = e^(2-i)x, i=sqrt(-1) are two linearly independent functions

hence obtain a second order linear differential equation with constant coefficients each that y1(x) and y2(x) are its two fundamental solutions.

my attempt :

for the first part, I use the definition of wroskian = y1y2'-y2y1' and show it not equal to zero... ok

the second part, I don't know how to do it...

how to get the second order differential equation?

is that setting : (r+2+i)(r+2-i) to get the auxillary equation? is that possible? can someone show me to solve this problem?

 

[g_edgar]

If you want that auxiliary equation, what will be the differential equation? When you get it, you can check whether the two original functions satisfy it.

 

[bobey]

how to do that... still blurr...