# order n constant coefficient differential equations and matrix exponential

Let $P$ be a degree $n>0$ monic complex polynomial in one indeterminate, let $f$ be a continuous function on the real line, let $k$ be an integer varying from 0 to $n-1$, and let $y_{k}$ be a complex number. The solution to the ODE

 $P(d/dt)\ y=f(t),\quad y^{(k)}(0)=y_{k}$ (1)

is

 $y(t)=\sum\ y_{k}\ g_{k}(t)+\int_{0}^{t}g_{n-1}(t-x)\ f(x)\ dx,$ (2)

where $g_{k}(t)$ is the coefficient of $z^{k}$ in the product of $P(z)$ by the singular part of

 $\frac{e^{tz}}{P(z)}\quad.$

Moreover, if $A$ is a complex square matrix annihilated by $P$, then

 $e^{tA}=\sum\ g_{k}(t)\ A^{k}.$ (3)

(1) into

 $Y^{\prime}-B\,Y=f(t)\ v,\quad Y(0)=Y_{0}$ (4)

by putting $Y_{k}:=y^{(k)}$, $Y_{0k}:=y_{k}$, and by letting $B$ be the transpose companion matrix of $P$, and $v$ the last vector of the canonical basis of $\mathbb{C}^{n}$. The solution to (4) is

 $Y(t)=e^{tB}\ Y_{0}+\int_{0}^{t}\ f(x)\ e^{(t-x)B}\ v\ dx.$

There is a unique $n$-tuple of functions $h_{k}$ such that $e^{tA}$ is the sum of the $h_{k}(t)\,A^{k}$ whenever $A$ is a complex square matrix annihilated by $P$. The first line of $B^{k}$ being the $(k+1)$-th vector of the canonical basis of $\mathbb{C}^{n}$ (for $0\leq k), we obtain

 $y(t)=\sum\ y_{k}\ h_{k}(t)+\int_{0}^{t}h_{n-1}(t-x)\ f(x)\ dx,$

so that the proof of (2) and (3) boils down to verifying

 $h_{k}(t)=g_{k}(t).$

a real value of $t$, let $G\in\mathbb{C}[X]$ be the sum of the $g_{k}(t)\,X^{k}$, form the entire function

 $\varphi(z)=\frac{e^{tz}-G(z)}{P(z)}\quad,$

multiply the above equality by $P(z)$, and replace $z$ by $A$.

Title order n constant coefficient differential equations and matrix exponential OrderNConstantCoefficientDifferentialEquationsAndMatrixExponential 2013-03-22 19:01:00 2013-03-22 19:01:00 gaillard (1824) gaillard (1824) 7 gaillard (1824) Definition msc 34-01