Simultaneous Homogeneous Linear First Order ODEs/Examples/y' - 3y + 2z = 0, y' + 4y - z = 0

Theorem
Consider the system of linear first order ordinary differential equations with constant coefficients:

The general solution to $(1)$ and $(2)$ consists of the linear combinations of the following:

Proof
Using the technique of Solution to Simultaneous Homogeneous Linear First Order ODEs with Constant Coefficients, we calculate the roots of the quadratic equation:


 * $\paren {k + a} \paren {k + d} - b c = 0$

where:

That is:
 * $\paren {k - 3} \paren {k - 1} - 8 = 0$

or:
 * $k^2 - 4 k - 5 = 0$

This has roots:

We also obtain:

When $k = 5$ we get that $A + B = 0$.

When $k = -1$ we get that $2 A - B = 0$.

This provides us with the solutions:

or:

From these, the general solution is constructed: