Showing the solution of an Euler DE oscillates

[Electroguru]

Homework Statement

I have the DE

which is an Euler DE. Show that if k >1/4, the solution of the DE would oscillate.

Homework Equations

e^ix= cos(x) +isin(x) I assume.

The Attempt at a Solution

I understand here that if k>1/4 the solution of the DE may oscillate, but if k

1/4, it will not. I understand why it would oscillate because the roots of the indicial equation would come on out as complex because if you were to plug the values into the quadratic formula, you would recieve a negative under the square root. Other than explaining it in this fashion, how would some one show this though?

Here I have r=[-1 +/- sqrt(1-4k)]/2

 

[Brian T]

Well if you have an imaginary characteristic root, then the solution contains cos(x) and sin(x) which would be oscillatory. If you want find the actual solutions in terms of k (assuming k > 1/4).