Linear System of ODE

Question

Would you mind telling me how do we use Matrix Algebra to get a general solution of the system:

$$x'=-{\delta}^{2}x+y+\delta z$$ $$y'=-x-{\delta}^{2}y+\delta z$$ $$z'=-\delta z$$

where $\delta$ is a parameter.

Answer 1

We rewrite this as:

$$x' = A x = \begin{bmatrix} -\delta^2 & 1 & \delta \\ -1 & -\delta^2 & \delta \\ 0 & 0 & -\delta \\ \end{bmatrix}x$$

We find the eigenvalues of this matrix using $|A - \lambda I| = 0$, which gives us:

$$-\delta - \delta^5 - \lambda - 2 \delta^3 \lambda - \delta^4 \lambda - \delta \lambda^2 - 2 \delta^2 \lambda^2 - \lambda^3 = -(\delta+\lambda) ((\delta^2+\lambda)^2+1) = 0$$

This gives us the three eigenvalues:

$$ \lambda_1 = -\delta,~ \lambda_2 = -\delta^2 + i, ~\lambda_3 = -\delta^2 - i$$

Now, you can find the eigenvectors and then write the solution for $x(t)$.

The three eigenvectors are:

• $v_1 = \left(\dfrac{\delta (\delta^2- \delta +1 )}{\delta^4-2 \delta^3 + \delta^2+1}, -\dfrac{-\delta^3 +\delta^2+\delta}{\delta^4-2 \delta^3 + \delta^2+1},1\right)$
• $v_2 = (-i, 1, 0)$
• $v_3 = (i, 1, 0)$

Also note, that $z$ is a decoupled equation and you can solve for it straight off, substitute back into the $x'$ and $y'$ and reduce this to a $2 x 2$ system, which is much easier to work with when dealing with parameters.

Answer 2

What you have here is a linear homogeneous system of ODEs with constant coefficients. Form the matrix $A$ given by \begin{array}{ccc} -\delta^2 & 1 & \delta\\ -1 & -\delta^2 & \delta\\ 0 & 0 & -\delta \end{array} and find the eigenvalues of $A$ as the roots of characteristic polynomial of $A$ (i.e. the determinant of the matrix $xI-A$). For each eigenvalue you will have to find a corresponding eigenvector. (In general you'll need to find a basis for the complete eigenspace, and in some cases generalized eigenvectors, but in your case the multiplicities are all one, so you'll just need a single eigenvector for each eigenvalue.) The solution to the system can then be expressed in terms of the eigenvalues and eigenvectors. (See your textbook for the formulas, but be careful, as two of the eigenvalues in this example may very well be complex.) Hope this is enough of a hint.