A first order lineal equation has the form where $a$ and $b$ are continuous functionsfrom an interval $[α, β]$ to the real line.

$b$ is called the inhomogeneity of the problem, and the equation where $b = 0$ is called the associated homogeneous equation. $u^{'} = a (t) u$

A solution of a first order linear equation is a $C^{1}$ function from $[α, β]$ to the real line such that the equation is satisfied at all times. We will denote the set of solutions of an equation with inhomogeneity $b$ as $Σ_{b}$ , and the solutions of the associated homogeneous system as $Σ_{0}$ .

Properties of the space of solutions

$Σ_{0}$ is a vector space; that is, it satifies the principle of superposition: linear combinations of solutions are solutions.

$Σ_{b}$ is an affine_space parallel to $Σ_{0}$ . That is, it satifies that the difference of any two solutions are in $Σ_{0}$ , and any element in $Σ_{0}$ plus other element in $Σ_{b}$ is an element from $Σ_{b}$ .

First order linear equations of constant coefficients

One special kind of linear equations are those in which the coefficients $a$ and $b$ are constant numbers Such linear equations are always resoluble. $u^{'} = a u + b$

To solve them, we first have to solve the associated homogeneous equation $u^{'} = a u$ .

This has as a solution the functions $k e^{\int_{t_{0}}^{t} a}$ for $k$ constant and $t_{0} \in [α, β]$ .

We can find a concrete solution of the inhomogeneous equation using variation of coefficients. We consider as a candidate to a solution the function $u = h ˙ v$ , for $h$ a solution of the homogeneous system such as $e^{\int_{t_{0}}^{t} a}$ .

Then if we plug $u$ into the equation we find that $u^{'} = (h v)^{'} = h^{'} v + h v^{'} = a u + b = a (h v) + b$ Since $h \in Σ_{0}$ , $h^{'} = a h$ , thus $v^{'} = b h^{- 1} = b e^{- \int_{t_{0}}^{t} a}$ Therefore we can integrate and we arrive to: $v = \int_{t_{0}}^{t} b e^{\int_{t}^{s} a} d s$ By the affinity of $Σ_{b}$ , we can parametrize it by $k e^{\int_{t_{0}}^{t} a} + \int_{t_{0}}^{t} b e^{\int_{t}^{s} a} d s$ for $k$ constant.