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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)=c_{1}e^{m1t}+c_{2}e^{m2t}

2) y(t)=c_{1}e^{mt}+c_{2}te^{mt}

3) y(t)=c_{1}e^{u}cos(v)+c_{2}sin(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,

m_{1}=-3 and m_{2}=-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 m_{1}& m_{2}?

Thanks for your contribution.

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