Help with initial value problem(IVP)

[shayaan_musta]

Hello experts!

As we know that there are 3 different general solutions of an ordinary differential equation depends on that what type of roots we've gotten in the solution, listed below.
1) y(t)=c₁e^m₁t+c₂e^m₂t
2) y(t)=c₁e^mt+c₂te^mt
3) y(t)=c₁e^ucos(v)+c₂sin(v)


Now 1st solution is used when discriminant i.e. D>0, 2nd is used when D=0 and 3rd is used when D<0

But here is a question,
y^''+6y^'+9y=0
By solving we get roots,
m₁=-3 and m₂=-3
While discriminant is -30
& book has used 2nd solution to solve it, if D<0 it must use 3rd solution.

How do we choose correct solution? Does solution is according to nature of roots(discriminant) or m₁ & m₂?

Thanks for your contribution.

 

[omer21]

Discriminant is not -30, it is 6^2-4.1.9=0

 

[shayaan_musta]

Ok.
Thanks to light me the right way.