Solution to Simultaneous Homogeneous Linear First Order ODEs with Constant Coefficients

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:

and:

where $A_1 : B_1 = A_2 : B_2 = r$

where $r$ is either of the roots of the quadratic equation:


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

Proof
We look for solutions to $(1)$ and $(2)$ of the form:

We do of course have the Trivial Solution of Homogeneous Linear 1st Order ODE:


 * $y = z = 0$

which happens when $A = B = 0$.

So let us investigate solutions where either or both of $A$ and $B$ are non-zero.

Substituting $(3)$ and $(4)$ into $(1)$ and $(2)$ and cancelling $e^{k x}$, we get::

From $(5)$ and $(6)$ we get:

So $A = B = 0$ unless $k$ is a root of the quadratic equation:


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

That is:


 * $(8): \quad \begin {vmatrix} k + a & b \\ c & k + d \end {vmatrix} = 0$

where the above notation denotes the determinant.

Assume $(8)$ has distinct roots $k_1$ and $k_2$.

Taking $k = k_1$ and $k = k_2$ in $(7)$, we can obtain ratios $A_1 : B_1$ and $A_2 : B_2$ such that:

and:

are solutions of $(1)$ and $(2)$.

By taking arbitrary linear combinations of these, we obtain the general solution.