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,
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{\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
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{\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
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{\displaystyle y_{t+n}=b_{1}y_{t+n-1}+\cdots +b_{n}y_{t}}
has characteristic equation
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{\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.
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...
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...
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...
Homework Statement
The stability of a spinning body may be explored by using equation (3.40), with no
torque components present. It will be assumed here that the spin is about the z -axis and
has a rate ωZ = S.
Homework Equations
$$I_{xx}\dot{ω} - (I_{yy}-I_{zz})Sω_y = 0$$
$$I_{yy}\dot{y} -...
Homework Statement
x2 d2y/dx2 + 3x dy/dx + 5y = g(x)Homework Equations
How do we find Characteristic equation for it.
The Attempt at a Solution
x2λ2 + 3xλ + 5 = 0
λ1 = 1/2 [-x2 + √ (x4 + 20 ) ]
λ2 = 1/2[ -x2 - √(x4 + 20) ]
I used 1/3 -/+ a √(a2 + 4b)
where
a = x2
b = 5
Homework Statement
The system is a spring with constant 3k hanging from a ceiling with a mass m attached to it, then attached to that mass another spring with constant 2k and another mass m attached to that.
So spring -> mass -> spring ->mass.
Find the normal modes and characteristic system...
An ODE of second order with constants coefficients, linear and homogeneous: Af''(x) + Bf'(x) +Cf(x) = 0 has how caractherisc equation this equation here: Ax^2 + Bx +C = 0 and has how solution this equation here: f(x) = a \exp(u x) + b \exp(v x) where u and v are the solutions (roots) of the...
Homework Statement
Regarding the case where the auxillary (characteristic) equation has complex roots, we solve the quadratic in the usual way using i to get the general solution
y(x) = e^{\alpha x}\left(C_1 \cos{\beta x} + i C_2 \sin{\beta x}\right)
And the textbook shows
y(x) = e^{\alpha...
Homework Statement
If I have the characteristic equation:
-λ3 + 3λ2 + 9λ + 5
And I'm told that one of its eigenvalues is -1.
How do I find the rest of the eigenvalues?
Homework Equations
-λ3 + 3λ2 + 9λ + 5
The Attempt at a Solution
The furthest I can get is:
-λ3 + 3λ2 + 9λ + 5 = (λ + 1) x...
Homework Statement
Given an NxN symetric tri-diagonal matrix, derive the recursion relation for the characteristic polynomial Pn(λ)
Homework Equations
Pn(λ) = |A -λI |
Pn(λ) = (An,n - λ)Pn-1(λ) - A2n,n-1Pn-2(λ)
The Attempt at a Solution
This was easy to do by induction, but I am always...
Any help is appreciated
1.)----Find the Character equation for the diff equation d^2y/dx^2-4dy/dx+3y=0 with initial conditions y(0)=0 and y'(0)=12 find the solution y(t)
(this is what I have gotten so far on this part) p^2+4p+3=0
then (p-1)(p-3)=0 so p1=1 and p2=3?
not really sure...
I am integrating the characteristic equation in order to recover the PMF, but I am going to get the answer to be zero so something went wrong.
\begin{align*}
p_X[k]...
Homework Statement
A:B→B a linear operator
Show r is multiple root for minimal polynomial u(x) iff
>$$\{0\}\subset \ker(A - rI) \subset \ker(A - rI)^2$$
note: it is proper subsetHomework Equations
The Attempt at a Solution
Homework Statement
My thought:
I know ker(A−rI) is basically {{0}...
Dear all,
Greetings! I was given a problem from Reichl's Statistical Physics book. Thank you very much for taking time to read my post.
Homework Statement
The stochastic variables X and Y are independent and Gaussian distributed with
first moment <x> = <y> = 0 and standard deviation...
Homework Statement
The stochastic variables X and Y are independent and Gaussian distributed with
first moment <x> = <y> = 0 and standard deviation σx = σy = 1. Find the characteristic function
for the random variable Z = X2+Y2, and compute the moments <z>, <z2> and <z3>. Find the first 3...
This was something I noticed as I was trying to practice solving PDEs using the method of characteristics.
The text has the following example: $$\frac{\partial u}{\partial x} + x \frac{\partial u}{\partial y} = 0$$
This should be easy enough. I let p(x,y) = x and solve for \frac{\partial...
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...
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...
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...
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 ?
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?
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...
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...
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
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...
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)?
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...
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...