First order linear equations

A first order lineal equation has the form $$u'=a(t)u+b(t)$$ where $a$ and $b$ are continuous functionsfrom an interval $[\alpha, \beta]$ 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 $[\alpha, \beta]$ 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 $\Sigma_b$, and the solutions of the associated homogeneous system as $\Sigma_0$.

Properties of the space of solutions

$\Sigma_0$ is a vector space; that is, it satifies the principle of superposition: linear combinations of solutions are solutions.

$\Sigma_b$ is an [-affine_space] parallel to $\Sigma_0$. That is, it satifies that the difference of any two solutions are in $\Sigma_0$, and any element in $\Sigma_0$ plus other element in $\Sigma_b$ is an element from $\Sigma_b$.

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. $$u' = au+b$$

To solve them, we first have to solve the associated homogeneous equation $u'=au$.

This has as a solution the functions $ke^{\int_{t_0}^ta}$ for $k$ constant and $t_0\in [\alpha, \beta]$.

We can find a concrete solution of the inhomogeneous equation using variation of coefficients. We consider as a candidate to a solution the function $u=h\dot v$, for $h$ a solution of the homogeneous system such as $e^{\int_{t_0}^ta}$.

Then if we plug $u$ into the equation we find that $$u'=(hv)'=h'v+hv'=au+b=a(hv)+b$$ Since $h\in\Sigma_0$, $h'=ah$, thus $$v'=bh^{-1}=be^{-\int_{t_0}^ta}$$ Therefore we can integrate and we arrive to: $$v=\int_{t_0}^tbe^{\int_{t}^sa}ds$$ By the affinity of $\Sigma_b$, we can parametrize it by $ke^{\int_{t_0}^ta}+\int_{t_0}^tbe^{\int_{t}^sa}ds$ for $k$ constant.