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How do I solve the following system of differential equations

  1. $x' = a_{1}x+b_{1}y+c_{1}z$;
  2. $y' = a_{2}x+b_{2}y+c_{2}z$;
  3. $z' = a_{3}x+b_{3}y+c_{3}z$.

Do I need any specific conditions here?

First, I guess a solution, $x(t) = Ae^{rt}$, $y(t) = Be^{rt}$, and $z(t) = Ce^{rt}$.

Then substitute the solution in the system of DEs:

  1. $rAe^{rt} = a_{1}Ae^{rt}+b_{1}Be^{rt}+c_{1}Ce^{rt}$;
  2. $rBe^{rt} = a_{2}Ae^{rt}+b_{2}Be^{rt}+c_{2}Ce^{rt}$;
  3. $rCe^{rt} = a_{3}Ae^{rt}+b_{3}Be^{rt}+c_{3}Ce^{rt}$.

Then the put the system of equations in a matrix:

$$\begin{bmatrix} a_{1}-r &b_{1} &c_{1} \\ a_{2}&b_{2}-1 &c_{2} \\ a_{3}&b_{3} &c_{3}-r \end{bmatrix} \begin{bmatrix} A\\ B\\ C \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}$$.

The determinant of the 3X3 matrix yields the characteristic equation and after skipping some steps, we get:

$$r^{3} - (a_{1}+b_{2}+c_{3})r^{2} - (-a_{1}b_{2}-a_{1}c_{3}-b_{2}c_{3}+b_{3}c_{2}+a_{3}c_{1}+a_{2}b_{1})r-a_{1}b_{2}c_{3}+a_{1}b_{3}c_{2}+a_{3}b_{2}c_{1}+a_{2}b_{1}c_{3}-a_{3}b_{1}c_{2}-a_{2}c_{1}b_{3} = 0.$$

Now, where do I go from here, or is there a simpler way of doing the problem?

EDIT: Tried doing with the eigenvalues and eigenvalues but get the same thing and can't solve for the eigenvalues.

OGC
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    I'd say you're supposed to give the answer as a function of the eigenvalues of the associated matrix (which you'll just label $\lambda_1, \lambda_2, \lambda_3$ or something). Otherwise this is hell. – Git Gud Feb 08 '15 at 19:20
  • @GitGud We weren't even taught about the eigenvalues. – OGC Feb 08 '15 at 19:29
  • @user36829: What techniques have you been taught? – science Feb 08 '15 at 19:31
  • Perhaps it's a prerequisite? They certainly help and I could go as far as saying they are necessary, but I won't. By the way, your guess at a solution is only right if there are three distinct eigenvalues, otherwise it won't work. – Git Gud Feb 08 '15 at 19:32
  • @science Well, it's not really a DE class per se. It's a first-year PhD economics class called Dynamic Optimization. And you know how economists do math ... we haven't been taught any formal techniques actually. Just guesswork to find the solution. – OGC Feb 08 '15 at 19:33
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    You may want to have a look at this. – science Feb 08 '15 at 19:45
  • Just correcting myself. I misread the exponents of $x(t), y(z), z(t)$. Your try at the moment will never work. – Git Gud Feb 08 '15 at 19:47
  • @GitGud I'm learning the method with the eigenvalues and will do it myself. Meanwhile feel free to post your solution if you want to. – OGC Feb 08 '15 at 20:03
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    I'm much more interested in the solution your instructor has in mind. In any case my solution would be a small generalization of this. – Git Gud Feb 08 '15 at 20:22
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    I'd just let $\mathbf{v}(t)= \exp(t\mathbf{A})\mathbf{v}(0)$, and leave it at that. – user14717 Feb 08 '15 at 20:42
  • I'll make an edit soon. – OGC Feb 08 '15 at 21:01
  • Even with the eigenvalues I am getting nowhere. – OGC Feb 08 '15 at 21:41
  • @user36829 What Nick said. See this. – Git Gud Feb 08 '15 at 22:01
  • @user36829 Yo. I want to help you, but I can't understand what kind of answer your instructor wants. An answer depending on $A$'s entries is completely unreasonable. And since you don't seem to have any tools are your disposal, we're left with nothing to go on.How do you usually solve systems of differential equations? – Git Gud Feb 09 '15 at 01:29

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