Characteristic equation

The characteristic equation is an aid in the theory of ordinary differential equations for calculating solutions of linear differential equations with constant coefficients. It reduces the determination of a fundamental system of the differential equation to the solution of a polynomial equation .

An analogous method can also be used to solve linear difference equations with constant coefficients.

Leonhard Euler reported on this method of solving differential equations in the case of constant coefficients in a letter to Johann I Bernoulli in 1739 , without considering multiple solutions of the characteristic equation. A solution for a differential equation with multiple zeros in the characteristic equation can be found later in Euler's Institutiones calculi integralis . Next have Augustin-Louis Cauchy and Gaspard Monge researched it.

definition

A homogeneous linear differential equation -th order of shape is given ${\displaystyle n}$

${\displaystyle a_{n}y^{(n)}(x)+a_{n-1}y^{(n-1)}(x)+\dotsb +a_{1}y'(x)+a_{0}y(x)=0}$

for a desired function with constant complex coefficients , , . Then the corresponding characteristic equation reads ${\displaystyle y\colon \mathbb {R} \to \mathbb {C} }$ ${\displaystyle a_{k}\in \mathbb {C} }$ ${\displaystyle k=0,\dotsc ,n}$ ${\displaystyle a_{n}\neq 0}$

${\displaystyle a_{n}\lambda ^{n}+a_{n-1}\lambda ^{n-1}+\dotsb +a_{1}\lambda +a_{0}=0}$.

The polynomial

${\displaystyle P(\lambda )=\sum _{k=0}^{n}a_{k}\lambda ^{k}}$

on the left side of the equation is also called the characteristic polynomial of the differential equation.

Formally, the characteristic equation is obtained by replacing the -th derivative of with the -th power of the polynomial variable (named here ). ${\displaystyle P(\lambda )=0}$ ${\displaystyle k}$ ${\displaystyle y}$ ${\displaystyle k}$ ${\displaystyle \lambda }$

solutions

According to the theory of linear differential equations, the solution set of a homogeneous linear differential equation -th order forms a -dimensional vector space . Accordingly, it is sufficient to find linearly independent solutions of the differential equation for the determination of the general solution . According to the fundamental theorem of algebra , the polynomial has precisely complex zeros if these are counted according to their multiplicity. In the following it is shown how with the help of these zeros of the characteristic equation a basis of the solution space of the differential equation, i.e. a fundamental system , can always be given. If there is such a basis in general , then is ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle P(\lambda )}$ ${\displaystyle n}$ ${\displaystyle \lambda _{1},\dotsc ,\lambda _{n}}$ ${\displaystyle \varphi _{1},\dotsc ,\varphi _{n}}$

${\displaystyle y(x)=c_{1}\varphi _{1}(x)+\dotsb +c_{n}\varphi _{n}(x)}$

with the general solution of the given differential equation. If there is an initial value or boundary value problem , the coefficients can then be determined from the additionally given conditions. ${\displaystyle c_{1},\dotsc ,c_{n}\in \mathbb {C} }$ ${\displaystyle c_{1},\dotsc ,c_{n}}$

Simple solutions

The approach with an unknown leads to the equation and thus to the characteristic equation after division . The following applies: ${\displaystyle y(x)=e^{\lambda x}}$ ${\displaystyle \lambda \in \mathbb {C} }$ ${\displaystyle y^{(k)}(x)=\lambda ^{k}e^{\lambda x}}$ ${\displaystyle \sum _{k=0}^{n}a_{k}\lambda ^{k}e^{\lambda x}=0}$ ${\displaystyle e^{\lambda x}\neq 0}$

The function is a solution of the differential equation if and only if is a solution of the characteristic equation. ${\displaystyle y(x)=e^{\lambda x}}$ ${\displaystyle \lambda }$

If all zeros are different from each other, you get different solutions of the differential equation in this way and it can be shown that these are also linearly independent. So the general solution in this case is ${\displaystyle \lambda _{1},\dotsc ,\lambda _{n}}$ ${\displaystyle n}$ ${\displaystyle y_{j}(x)=e^{\lambda _{j}x}}$

${\displaystyle y(x)=c_{1}e^{\lambda _{1}x}+\dotsb +c_{n}e^{\lambda _{n}x}}$

with freely selectable constants . ${\displaystyle c_{1},\dotsc ,c_{n}\in \mathbb {C} }$

Multiple solutions

If, on the other hand, there is a multiple solution of the characteristic equation, then only one solution is obtained in this way , i.e. no more fundamental system. In this case, however, further linearly independent solutions can be given in a simple way: ${\displaystyle \lambda }$ ${\displaystyle e^{\lambda x}}$

If there is a -fold zero of the characteristic polynomial, then are linearly independent solutions of the differential equation. ${\displaystyle \lambda }$ ${\displaystyle m}$ ${\displaystyle y_{1}(x)=e^{\lambda x},y_{2}(x)=xe^{\lambda x},\dotsc ,y_{m}(x)=x^{m-1}e^{\lambda x}}$

Complex solutions for a real equation

In the following all coefficients are real numbers . In this case one is often only interested in real solutions of the differential equation and thus also in a real fundamental system. If with , is a complex solution of , then the complex conjugate number is also a solution. These correspond to linearly independent complex solutions and the differential equation. With the help of Euler's formula one gets from this ${\displaystyle a_{0},\dotsc ,a_{n}}$ ${\displaystyle \lambda =\alpha +i\beta }$ ${\displaystyle \alpha ,\beta \in \mathbb {R} }$ ${\displaystyle \beta \neq 0}$ ${\displaystyle P(\lambda )=0}$ ${\displaystyle {\overline {\lambda }}=\alpha -i\beta }$ ${\displaystyle y_{+}(x)=e^{\lambda x}=e^{\alpha x}e^{i\beta x}}$ ${\displaystyle y_{-}(x)=e^{{\overline {\lambda }}x}=e^{\alpha x}e^{-i\beta x}}$

${\displaystyle y_{1}(x)={\frac {y_{+}(x)+y_{-}(x)}{2}}=\operatorname {Re} (y_{+}(x))=e^{\alpha x}\cos(\beta x)}$

and

${\displaystyle y_{2}(x)={\frac {y_{+}(x)-y_{-}(x)}{2i}}=\operatorname {Im} (y_{+}(x))=e^{\alpha x}\sin(\beta x)}$

as real solutions of the differential equation. These are also linearly independent. Similarly, in the case of multiple complex solutions, one can construct two linearly independent real solutions by transitioning to the real and imaginary part to each pair of conjugate complex solutions. The conjugate complex solutions result in the two real solutions and . ${\displaystyle x^{j}e^{(\alpha \pm i\beta )x}}$ ${\displaystyle x^{j}e^{\alpha x}\cos(\beta x)}$ ${\displaystyle x^{j}e^{\alpha x}\sin(\beta x)}$

Examples

• The characteristic equation of the differential equation is and has the solutions and . This gives the fundamental system , and the general solution of the differential equation is . ${\displaystyle y''+y'-6y=0}$ ${\displaystyle \lambda ^{2}+\lambda -6=0}$ ${\displaystyle \lambda _{1}=2}$ ${\displaystyle \lambda _{2}=-3}$ ${\displaystyle y_{1}(x)=e^{2x}}$ ${\displaystyle y_{2}(x)=e^{-3x}}$ ${\displaystyle y(x)=c_{1}e^{2x}+c_{2}e^{-3x}}$
• The oscillation equation with has the characteristic equation with the conjugate complex solutions . So a complex fundamental system is a real one . So the general solution is . ${\displaystyle {\ddot {x}}(t)+\omega _{0}^{2}x(t)=0}$ ${\displaystyle \omega _{0}>0}$ ${\displaystyle \lambda ^{2}+\omega _{0}^{2}=0}$ ${\displaystyle \lambda _{1,2}=\pm i\omega _{0}}$ ${\displaystyle e^{i\omega _{0}t},e^{-i\omega _{0}t}}$ ${\displaystyle \cos(\omega _{0}t),\sin(\omega _{0}t)}$ ${\displaystyle x(t)=c_{1}\cos(\omega _{0}t)+c_{2}\sin(\omega _{0}t)}$
• The differential equation

${\displaystyle {y}^{(7)}-16\,{y}^{(6)}+108\,{y}^{(5)}-392\,{y}^{(4)}+804\,{y'''}-880\,{y''}+400\,y'=0}$

has the characteristic equation

${\displaystyle {\lambda }^{7}-16\,{\lambda }^{6}+108\,{\lambda }^{5}-392\,{\lambda }^{4}+804\,{\lambda }^{3}-880\,{\lambda }^{2}+400\,\lambda =0}$.

This has the seven zeros (with multiplicity)

${\displaystyle 0,2,2,3+i,3+i,3-i,3-i}$.

This gives the real fundamental system

${\displaystyle 1,e^{2x},xe^{2x},e^{3x}\cos(x),e^{3x}\sin(x),xe^{3x}\cos(x),xe^{3x}\sin(x)}$

and the general solution

${\displaystyle y(x)=c_{1}+c_{2}e^{2x}+c_{3}xe^{2x}+c_{4}e^{3x}\cos(x)+c_{5}e^{3x}\sin(x)+c_{6}xe^{3x}\cos(x)+c_{7}xe^{3x}\sin(x)}$.

literature

• Herbert Amann: Ordinary differential equations , 2nd edition, Gruyter - de Gruyter textbooks, Berlin New York 1995, ISBN 3-11-014582-0 , section 14, pp. 205-217.

Individual evidence

1. ↑ Ilja Bronstein among other things: Pocket book of mathematics. 7th edition. Scientific publishing house Harri Deutsch, Frankfurt, 2008, ISBN 978-3-8171-2007-9 , p. 559 ( limited preview in the Google book search).
2. ↑ E863 - The correspondence between Leonhard Euler and Johann I Bernoulli, 1727–1740. Published z. B. in the third part of the series by G. Eneström, Bibl. Math. 63, 1905, p. 37.
3. ↑ Institutiones calculi integralis, 1768–1770, Part Two, Chapter 4, Problem 102.