# 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:

\(\text {(1)}: \quad\) | \(\ds \dfrac {\d y} {\d x} + a y + b z\) | \(=\) | \(\ds 0\) | |||||||||||

\(\text {(2)}: \quad\) | \(\ds \dfrac {\d x} {\d z} + c y + d z\) | \(=\) | \(\ds 0\) |

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

\(\ds y\) | \(=\) | \(\ds A_1 e^{k_1 x}\) | ||||||||||||

\(\ds z\) | \(=\) | \(\ds B_1 e^{k_1 x}\) |

and:

\(\ds y\) | \(=\) | \(\ds A_2 e^{k_2 x}\) | ||||||||||||

\(\ds z\) | \(=\) | \(\ds B_2 e^{k_2 x}\) |

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:

\(\text {(3)}: \quad\) | \(\ds y\) | \(=\) | \(\ds A e^{k x}\) | |||||||||||

\(\text {(4)}: \quad\) | \(\ds z\) | \(=\) | \(\ds B e^{k x}\) |

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::

\(\text {(5)}: \quad\) | \(\ds \paren {k + a} A + b B\) | \(=\) | \(\ds 0\) | |||||||||||

\(\text {(6)}: \quad\) | \(\ds c A + \paren {k + d} B\) | \(=\) | \(\ds 0\) |

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

\(\text {(7)}: \quad\) | \(\ds \) | \(\) | \(\ds \paren {\paren {k + a} \paren {k + d} - b c} A\) | |||||||||||

\(\ds \) | \(=\) | \(\ds \paren {\paren {k + a} \paren {k + d} - b c} B\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 0\) |

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:

\(\ds y\) | \(=\) | \(\ds A_1 e^{k_1 x}\) | ||||||||||||

\(\ds z\) | \(=\) | \(\ds B_1 e^{k_1 x}\) |

and:

\(\ds y\) | \(=\) | \(\ds A_2 e^{k_2 x}\) | ||||||||||||

\(\ds z\) | \(=\) | \(\ds B_2 e^{k_2 x}\) |

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

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

This needs considerable tedious hard slog to complete it.In particular: Cover the case where $k_1 = k_2$. The source work is vague on this subject. Recommend this solution be reworked, preferably in conjunction with a more rigorous and thorough source work than the one used here.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1958: G.E.H. Reuter:
*Elementary Differential Equations & Operators*... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 3$. Equations of higher order and systems of first order equations: $\S 3.2$ First order systems