# What is Characteristic equation: Definition and 30 Discussions

In mathematics, the characteristic equation (or auxiliary equation) is an algebraic equation of degree n upon which depends the solution of a given nth-order differential equation or difference equation. The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. Such a differential equation, with y as the dependent variable, superscript (n) denoting nth-derivative, and an, an − 1, ..., a1, a0 as constants,

a

n

y

(
n
)

+

a

n

1

y

(
n

1
)

+

+

a

1

y

+

a

0

y
=
0
,

{\displaystyle a_{n}y^{(n)}+a_{n-1}y^{(n-1)}+\cdots +a_{1}y'+a_{0}y=0,}
will have a characteristic equation of the form

a

n

r

n

+

a

n

1

r

n

1

+

+

a

1

r
+

a

0

=
0

{\displaystyle a_{n}r^{n}+a_{n-1}r^{n-1}+\cdots +a_{1}r+a_{0}=0}
whose solutions r1, r2, ..., rn are the roots from which the general solution can be formed. Analogously, a linear difference equation of the form

y

t
+
n

=

b

1

y

t
+
n

1

+

+

b

n

y

t

{\displaystyle y_{t+n}=b_{1}y_{t+n-1}+\cdots +b_{n}y_{t}}
has characteristic equation

r

n

b

1

r

n

1

b

n

=
0
,

{\displaystyle r^{n}-b_{1}r^{n-1}-\cdots -b_{n}=0,}
discussed in more detail at Linear difference equation#Solution of homogeneous case.
The characteristic roots (roots of the characteristic equation) also provide qualitative information about the behavior of the variable whose evolution is described by the dynamic equation. For a differential equation parameterized on time, the variable's evolution is stable if and only if the real part of each root is negative. For difference equations, there is stability if and only if the modulus (absolute value) of each root is less than 1. For both types of equation, persistent fluctuations occur if there is at least one pair of complex roots.
The method of integrating linear ordinary differential equations with constant coefficients was discovered by Leonhard Euler, who found that the solutions depended on an algebraic 'characteristic' equation. The qualities of the Euler's characteristic equation were later considered in greater detail by French mathematicians Augustin-Louis Cauchy and Gaspard Monge.

View More On Wikipedia.org
1. ### Engineering Ordinary Diferential Equations in electric circuit

It's a multiple choice exercise and I have managed to find the characteristic equation V0(t) which is ##V_0(t)= C_1e^{-t}+C_2e^{-3t}## Initially I thought that it was a non homogeneous ODE, but doing the math for the right part of the equation, I found out that it equals to 0. So, I need help...
2. ### Repeated roots of a characteristic equation of third order ODE

The characteristic equation ## m^3 -6m^2 + 12m -8 = 0## has just one single, I mean all three are equal, root ##m=2##. So, one of the particular solution is ##y_1 = e^{2x}##. How can we find the other two? The technique ##y_2 = u(x) e^{2x}## doesn't seem to work, and even if it were to work how...
3. ### MHB Solving Matrix A: Characteristic Equation and Eigenvectors

good evening everyone! Decided to solve the problems from last year's exams. I came across this example. Honestly, I didn't understand it. Who can help a young student? :) Find characteristic equation of the matrix A in the form of the polynomial of degree of 3 (you do not need to find...

17. ### IV characteristic equation of triple junction solar cell

Homework Statement The given quantities are the shunt resistances across each of the 3 diode junctions,assume them to be Rsh1,Rsh2,Rsh3; Tunnel diode resistances as Rt1 and Rt2, Photocurrent for each of the junction be Ip1,Ip2,Ip3 and bandgap of each subcell be Eg1,Eg2,Eg3 are given.Let V & I...
18. ### Complex root for characteristic equation

Suppose your characteristic equation for the 2nd order equation has complex roots r+ and r- These are conjuagtes of each other so the general solution is: y = Aer+ + Ber- My book chooses the constants A and B as conjugates of each other for the reason that this constructs a real...
19. ### 2nd order characteristic equation standart form

A second order system has the following standart form; http://controls-design.com/mathtex/mathtex.cgi?H%28s%29%3DK%5Cfrac%7B%5Comega_n%5E2%7D%7Bs%5E2%2B2%5Czeta%5Comega_n%20s%2B%5Comega_n%5E2%7D%20%5Cmbox%7B%20for%20%7D%200%20%5Cle%20%5Czeta%20%5Cle%201 However, sometimes the system I...
20. ### Characteristic equation formula for a nxn matrix ?

So I know that the characteristic equation for a 2x2 matrix can be given by t^2 - traceA + |A| So how would this be generalised for a 4x4 or higher matrix ?
21. ### Coefficients of the characteristic equation

let's say you are given the following matrix T: T11 T12 . . . T1n T21 T22 . . . T2n . . . . . . . ...
22. ### Solving Characteristic Equation: y'''-y''+y'-y=0

I am stuck on solving for the roots of a charactristic equation: y'''- y''+y'-y=0 where I set r^3-r^2+r-1=0 and factored out r to get r*[ r^2-r +1] -1 =0 to get the real root of 1. How can I solve for the compex roots?
23. ### Characteristic equation of binomial random variable

Homework Statement find the characteristic equation of a binomial variable with pmf p(x) =\frac{n!}{(n-k)!k!}*p^{k}*(1-p)^{n-k}Homework Equations characteristic equation I(t) = \sump(x)*e^{tk}The Attempt at a Solution I(t) = \sum\frac{n!}{(n-k)!k!}*(p^{k}*(1-p)^{-k}*e^{tk})*(1-p)^{n} i am...
24. ### HELP Characteristic Equation question

Homework Statement Come up with the frequency directly from the solutions of the characteristic equation. {{z=0.-5.71839 i},{z=0.+5.71839 i}} Homework Equations characteristic equation = z^2+b z+c=0 The Attempt at a Solution Not sure where to start. Any help would be greatly...
25. ### A question about the characteristic equation

Hi PF readers, When trying to establish \lambda values by solving a characteristic equation (for simplicity of 2x2 matrix) can one produce solution that contains complex roots? If yes, what does that show about the eigenvectors? Thanks in advance! Cygni
26. ### Matlab - ODE, find roots of the characteristic equation for the natural response

Homework Statement I need to use MATLAB to solve these problems. http://users.bigpond.net.au/exidez/IVDP.jpg Homework Equations MATLAB The Attempt at a Solution a) R1=3.6; R2=R1; C1=33*10^-6; C2=22*10^-6; % defining the polynomial constants Vs=[R1*R2*C1*C2...
27. ### Characteristic Equation for Asin(Wt) + Bcos(Wt): Explained

http://users.on.net/~rohanlal/qM.jpg I don't understand how this answer is obtained for the homogenous solution. What does characteristic equation in "r" mean and how does it help achieve the final solution of Asin(Wt) + Bcos(Wt)?
28. ### Characteristic equation

For the ODE x''+2x'+6x=sin2t what is the characteristic equation? Is it m^2+2m+6=0 or m^2+2m+6=cos2t
29. ### Characteristic Equation for Parallel Circuit with Current Source and 3 Elements

Homework Statement im trying to find the characteristic equation of a circuit with a current source and 3 elements all in parallel: a resistor and 2 inductors L1 and L2. Homework Equations i believe the current can be calculated as: i(t) = v(t)/R + iL1(t) + iL2(t) The Attempt at a...
30. ### Characteristic equation

Let's say I'm given a DEQ: (1) y^{(n)}+a_{n-1}\cdot y^{(n-1)}+\ldots + a_{0}\cdot y=0, where y is a real function of the real variable t, for example. I could now rewrite this as a system of DEQ in matrix form (let's not discuss why I would do that): (2) x'=Ax,\quad x=(y,\ldots,y^{(n-1)}). If I...