First order linear equations

A first order lineal equation has the form $$u^{\prime }=a\left(t\right)u+b\left(t\right)$$ 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. $$u^{\prime }=a\left(t\right)u$$

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. $$u^{\prime }=au+b$$

To solve them, we first have to solve the associated homogeneous equation $u^{\prime }=au$.

This has as a solution the functions $k{e}^{{\int }_{t_0}^{t}a}$ 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 $e^{{\int }_{t_0}^{t}a}$.

Then if we plug $u$ into the equation we find that $$u^{\prime }=(hv{)}^{\prime }=h^{\prime }v+h{v}^{\prime }=au+b=a(hv)+b$$ Since $h\in {\Sigma }_0$, $h^{\prime }=ah$, thus $$v^{\prime }=b{h}^{-1}=b{e}^{-{\int }_{t_0}^{t}a}$$ Therefore we can integrate and we arrive to: $$v={\int }_{t_0}^{t}b{e}^{{\int }_t^{s}a}ds$$ By the affinity of ${\Sigma }_b$, we can parametrize it by $k{e}^{{\int }_{t_0}^{t}a}+{\int }_{t_0}^{t}b{e}^{{\int }_t^{s}a}ds$ for $k$ constant.