In each of 1-4 find the general solution of the given system of differential equations

Chapter 3, Problem 2

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QUESTION:

In each of 1-4 find the general solution of the given system of differential equations.

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QUESTION:

In each of 1-4 find the general solution of the given system of differential equations.

ANSWER:

Step 1 of 6

 

Determine the eigenvalues of the matrix A.

\(\begin{aligned} \operatorname{det}(\mathbf{A}-\lambda \mathbf{I}) & =0 \\ \left|\begin{array}{ccc} -\lambda & -1 & 1 \\ 2 & -3-\lambda & 1 \\ 1 & -1 & -1-\lambda \end{array}\right| & =0 \\ -\lambda[(-3-\lambda)(-1-\lambda)-(-1)]-(-1)[2(-1-\lambda)-1]+1[-2-(-3-\lambda)] & =0 \\ -\lambda\left(3+3 \lambda+\lambda+\lambda^{2}+1\right)+(-2-2 \lambda-1)+(-2+3+\lambda) & =0 \\ -3 \lambda-3 \lambda^{2}-\lambda^{2}-\lambda^{3}-\lambda-2-2 \lambda-1-2+3+\lambda & =0 \\ -\lambda^{3}-4 \lambda^{2}-5 \lambda-2 & =0 \\ \lambda^{3}+4 \lambda^{2}+5 \lambda+2 & =0 \\ (\lambda+1)\left(\lambda^{2}+3 \lambda+2\right) & =0 \\ (\lambda+1)(\lambda+2)(\lambda+1) & =0 \\ (\lambda+2)(\lambda+1)^{2} & =0 \\ \lambda_{1}=-2 \text { or } \lambda_{2}=-1 \text { or } \lambda_{3} & =-1 \end{aligned}\)

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