Linear Systems of Differential Equations

When working with linear systems of equations that have the following form:

One can create a matrix system like the following:

One then can use differential equations - matrix operations to continue your work. See 1-Review of Matrix Algebra for more info.

When working with an equation that include derivatives of a higher order than just one you can employ the following method of decomposition to create a first order linear system.

First establish vector as the following:

Which now means that we now can write our system of equations in terms of instead of . This in-turn changes how we decompose the matrix specified in the First Order Linear Systems section above.

#incomplete

See 1-Review of Matrix Algebra for more info.

To solve an initial value problem with a linear system of differential equations do the following:

1. Put the system into matrix form (as specified in the section above).
2. Compute the eigenpairs - ( for eigenvalues and for eigenvectors).
3. Using the eigenpairs build the fundamental matrix, :

expand and check

When working with some matrices you may end up with an imaginary number when trying to find the eigenpairs. In this situation you can use complex eigenpairs and Euler's Formula to solve.

See nonhomogeneous linear systems for info about nonhomogeneous systems.

When working with a linear system of differential equations one has two main theorems to prove that a system is exists and is unique for some interval. Which theorem one uses depends on whether or not the system is linear.

When drawing a linear system of differential equations on a direction field the eigenvectors of a linear system are phase-plane lines that will pass through the origin of the xy-graph. This is because of a fundamental property of eigenvectors, see eigenpairs with differential equations for more info. The slope of the phase-plane lines given by a line though the origin and the eigenvectors as defined below:

The slope of the arrows along these lines align with the line itself. The direction is given by the eigenvalue of the eigenvector in question. Negative eigenvalues converge to the origin of the xy graph while positive eigenvalues diverge.

For arrows that are not on one of the eigenvectors given in the system one must factor in the arrow's proximity to the eigenvectors, the closer the arrow is to an eigenvector the more the arrow aligns with the arrows along the eigenvector.

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Thus one can plot the direction field of a linear system of differential equations by first plotting the eigenvectors of the system then using the directions provided by the eigenvalues plot the actual direction arrows.

See the following for more info:

• Elementary Differential Equations - Kohler & Johnson - Second Edition - Pg. 406
• 1-Sects 6.2 and 6.6 Stability

Defective Matrices