# 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

$$P(d/dt)y=f(t),{y}^{(k)}(0)={y}_{k}$$ | (1) |

is

$$y(t)=\sum {y}_{k}{g}_{k}(t)+{\int}_{0}^{t}{g}_{n-1}(t-x)f(x)\mathit{d}x,$$ | (2) |

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

$$\frac{{e}^{tz}}{P(z)}\mathit{\hspace{1em}}.$$ |

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

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

(1) into

$${Y}^{\prime}-BY=f(t)v,Y(0)={Y}_{0}$$ | (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 ${\u2102}^{n}$. The solution to (4) is

$$Y(t)={e}^{tB}{Y}_{0}+{\int}_{0}^{t}f(x){e}^{(t-x)B}v\mathit{d}x.$$ |

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 ${\u2102}^{n}$ (for $$), we obtain

$$y(t)=\sum {y}_{k}{h}_{k}(t)+{\int}_{0}^{t}{h}_{n-1}(t-x)f(x)\mathit{d}x,$$ |

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

$${h}_{k}(t)={g}_{k}(t).$$ |

a real value of $t$, let $G\in \u2102[X]$ be the sum of the ${g}_{k}(t){X}^{k}$, form the entire function^{}

$$\phi (z)=\frac{{e}^{tz}-G(z)}{P(z)}\mathit{\hspace{1em}},$$ |

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

Title | order n constant coefficient differential equations and matrix exponential |
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Canonical name | OrderNConstantCoefficientDifferentialEquationsAndMatrixExponential |

Date of creation | 2013-03-22 19:01:00 |

Last modified on | 2013-03-22 19:01:00 |

Owner | gaillard (1824) |

Last modified by | gaillard (1824) |

Numerical id | 7 |

Author | gaillard (1824) |

Entry type | Definition |

Classification | msc 34-01 |