Quadratic Solution to Homogeneous Second-Order ODEs

[epsilonzero]

Homework Statement

Consider ax''+bx'+cx=0 for b^2-4ac=0 with k= -b/(2a).

Show x=e^(kt) is a solution.

The Attempt at a Solution

x=e^(kt)
x'=(-b/2a)e^((-b/2a)t)
x''=(-b^2/4a^2)e^((-b/2a)t)

Then i plugged these into ax''+bx'+cx=0 and simplified to get e^((-b/2a)t)(-((b^2)/2)+c)=0

I'm stuck at what to do next.

Thank you for any help.

 

[Defennder]

To make things easier, don't use the k=-b/2a substitution until you're ready to compute the final answer. This helps to keep it a lot more tidier. Just subsitute the derivatives of x into the DE, factor out e^kt and make use of k = -b/2a as well as b^2 - 4ac = 0. It'll all cancel out to give 0.

 

[epsilonzero]

Ok, just got it. Thanks.