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.