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Question

Solve the following linear, homogeneous system of ODEs:

    

the 3 eigenvalues of the coefficient matrix are: 

Answer 1

Knowing the eigenvalues is not enough in this case, because you have repeated eigenvalues. You need to find the Jordan canonical form of the coefficient matrix. I assume you learned it already.

Let us call the matrix

|6 4 3|

|-4 -2 -3|

|3 3 5|

by A. The Jordan canonical form of A is

|2 1 0|

J = |0 2 0|,

|0 0 5|

with the transformation matrix

|-1 -1 1|

S = |1 0 -1|

|0 1 1|. We have A=SJS^-1.

We will have new variables u=S^-1x, namely

u₁ = -x₁-x₂+x₃

u₂ = x₁+0-x₃

u₃ = 0+x₂+x_3.

Under the new variable, the system is

u₁' = 2u₁ + u₂

u₂' = 2u₂

u₃' = 5u₃.

We have solutions in terms of new variables,

u₁ = te^2t

u₂ = e^2t

u₃ = e^5t.

Now all you need to do is to rewrite the above solution in terms of x's, which is x = Su.