Characteristic equation Definition and 30 Threads

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 ?

let's say you are given the following matrix T: T11 T12 . . . T1n T21 T22 . . . T2n . . . . . . . ...

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)?

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

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...