2nd order Definition and 494 Threads

1) "Undamped system is forced at the same frequency as one of its natural frequencies." Consider the 2nd order differential equation $$\ddot{x}+\omega_0^2x=F_0\cos{\omega t}\tag{1}$$ which models a mass attached to a spring (attached to a wall) with spring constant ##k## and...

As we noted above, stability is all about the solution to the homogeneous equation. For the equation $$y''+by'+ay=0\tag{3}$$ we have discriminant $$\Delta = b^2-4a\tag{4}$$ and the roots are $$r=\frac{-b\pm\sqrt{b^2-4a}}{2}\tag{5}$$ We have three cases. Case 1 (Distinct Real Non-Complex...

I have attached my attempt at a solution. In the solution image, I have computed 3 things: 1. System transfer function based on my understanding of the problem statement. This is a 2nd order system with steady state dc gain=0.9. So I wrote the transfer function accordingly. However, I strongly...

hello i have a question about kinetics : to have the integrated rate law for second order reaction the professor write the following why we don't write the rate like this : rate = -1/2(d[1]/dt) ? why we ignore the stoichiometric coefficient ?

Equation: , where matrix D, C, G and F can be represented by I'm supposed to design a control system that looks like this: I am given that the dynamic model = fcn(D,C,G,dq) where the dq is the same as 𝑞̇ and d2q in the diagram is the same 𝑞̈. The default initial value of [𝑞(0), 𝑞̇(0)] is...

The top most 2nd order non-linear DE is the one that has to be solved. Below is the solution. This problem is from Morin's Classical Mechanics. May I know how he could guess that r = Agt^2? Firstly, why must g tilda be a variable within r? I do not understand what he meant by 'parameter'...

Hello everyone, I am struggling with calculating the coefficients for second order transient analysis. For example, when analyzing a underdamped circuit, we know that the equation for voltage or current is xt=e-αt(K1cos(sqrt(ω2-α2)t ) + K2sin(sqrt(ω2-α2)t)). Then in order to determine for...

This is a physics problem from Griffith's Electrodynamics. I'm mainly asking about the math here. I found the DE in the box at part (d). To solve it, I did: ##\sqrt V {d^2 V} = \beta dx^2## Integrating twice: ##\frac {4} {15} V^{2.5} = \beta x^2/2## Why is my method wrong? Thanks for the help.

I just want to make sure I am on the right track here (hence have not given the other information in the question). In taking the Fourier transform of the PDE above, I get: F{uxx} = iω^2*F{u}, F{uxt} = d/dt F{ux} = iω d/dt F{u} F{utt} = d^2/dt^2 F{u} Together the transformed PDE gives a second...

Hi guys, I have just started studying DEs on my own, so pardonne moi in advance for the probably silly question :) Via Newton's second law of motion: $$x''=\frac{F}{m} \ [1]$$ Which is a second-order differential equation. But, from here, how do I get the good old equation of motion...

Consider the gaussian kick potential, ##\hat{V}(t) = \hat{x} \exp{(\frac{-t^2}{2 \tau^2})}## where ##\hat{x} = a+a^\dagger## in terms of creation and annihilation operators. Then we define the potential in the interaction picture, ##\hat{V}_I(t) = e^{i\hat{H}t}\hat{V}(t)e^{-i\hat{H}t}## I...

This is a very simple question: I would like to solve for ##\psi## in this equation $$\frac{d^{2}\psi}{d\xi^2} =\xi^2\psi$$ I so apply ##y=c_{1}e^{-kx}+c_{2}e^{kx}## and ##\psi## should be equal to ##\psi=c_{1}e^{-\xi^2}+c_{2}e^{\xi^2}##, because ##(D^2-\xi^2)\psi=0##. However the answer is...

Hi! I am looking into a mechanical problem which reduces to the set of PDE's below. I would be very happy if you could help me with it. I have the following set of second order PDE's that I want to solve. I want to solve for the generic solutions of the functions u(x,y) and v(x,y). A, B and C...

So here's my homework question: This is the reference formula along with the Rung-Kutta form with the variables mentioned in the question Here is my attempt so far: Problem is that i am unsure how to expand this to even get going. I tried referencing my text Math Methods by Boas which has...

Actually I was trying to write a small program in Scilab to simulate a quantum particle. When I give a potential higher than energy, the wave function should go like exp(-x) and decay. But my program just increases without bound. Is there any nice way to do anything about it?

In the following I will try to deduce the scattering amplitude for a specific interaction. My question is at the bottom, the entire rest is my reasoning to explain how I came to the results I present. My working Let's assume I would like to calculate the second order scattering amplitude in ##...

Hi everyone, I'm struggling with the proof for the second order energy correction for perturbation theory when substituting in the first order wavefunction. I have attached an image of my current proof for it below, but I'm not sure whether this is the correct approach for it (the H's in the...

y''(x)+y'(x)+F(x)=0 Pleas me a idea

Here is my problem I have given this a go and get 26.77 degrees as my angular position My concern is do I double this angle to get the angular width between both 2nd order maxima's (which would be 53.53 degrees) or do I just leave it as 26.77 degrees? Thanks for any help!

In second order case we should rewrite the equation in terms of 2 first order DE's. So I wrote, $$dx/dt = wx$$ $$dwx/dt = -GMx/r^3$$ and $$dy/dt = wy$$, $$dwy/dt = -GMy/r^3$$ Now I guess there's two ways to do it in 4th order RK method. I would either do it component by component or just in...

I have taken a look but most books and Online stuff just menctions the First order Taylor for 3 variables or the 2nd order Taylor series for just 2 variables. Could you please tell me which is the general expression for 2nd order Taylor series in 3 or more variables? Because I have not found...

2000 Convert the differential equation $$\displaystyle y^{\prime\prime} + 5y^\prime + 6y =0$$ ok I presume this means to find a general solution so $$\lambda^2+5\lambda+6=(\lambda+3)(\lambda+2)=0$$ then the roots are $$-3,-2$$ thus solutions $$e^{-3x},e^{-2x}$$ ok I think the Wronskain...

given the differential equation $\quad y''+5y'+6y=0$ (a)convert into a system of first order (homogeneous) differential equation (b)solve the system. ok just look at an example the first step would be $\quad u=y'$ then $\quad u'+5u+6=0$ so far perhaps?

It is true that at resonance frequency the phase-shift between input and output is 90 degrees, so my mind would think that this is ok. But I am kind of unsure because of the whole dividing by zero part. If this isn't allowed: is there any way to calculate/measure the damping coefficient with...

I want to solve ##\frac{du^2}{d\theta ^2}+u=\frac{GM}{h^2}## for ##u(\theta)##, where ##\frac{GM}{h^2}=constant##. The given equation is a nonhomogeneous second order linear DE. I begin by solving the associated homogeneous DE with constant coefficients: ##\frac{du^2}{d\theta ^2}+u=0## which...

I was stuck with the last line of my solution please help me find the correct answer to this problem. TIA!

Homework Statement Show that the 2nd order nondegenerate perturbation theory corrections are given by: ##E_n^2 = \sum_{k \neq n}^{\infty} \frac{|\left < \phi_n | \hat{H} | \phi_k \right> |^2}{E_n^0 - E_k^0}##[/B] and ## C_{nm}^2 = \frac{C_{nm}^1 E_n^1 - \sum_{k \neq n}^{\infty} C_{nk}^1...

Hi There is an example in my textbook worded as follows; A particle of mass 2kg moves along the positive x-axis under the action of a force directed towards the origin. At time t seconds, the displacement of P from O is x metres and P is moving away from O with a speed of v ms^-1. The force has...

Hi, I’m just wondering about this: Are there any theoretical reasons why physical laws take the form of 2nd order (in time) differential equations? Or is it just observed to be that way? Are there ANY laws (even in a limited context) which are 3rd (or higher) order in time?

Hello, $\vec{x'}=\small\begin{pmatrix}1&2\\3&2\end{pmatrix}\vec{x}+t\small\begin{pmatrix}2\\-4\end{pmatrix}$ Now i got the solution to this differential equation system as...

My function: d2f/dx2 + cf = delta(x) Condition: f is finite and f(50) = f(-50) = 0 Solution: f = C1exp(cx) + C2exp(-cx) Due to condition, f = C1exp(cx) for x<=0 and C2exp(-cx) for x>=0 f(50) = C2exp(-c*50) = 0 = > C2 = 0 Likewise, for C1 I don't know if I might have missed something...

Hi, I'm dealing with the following problem. I hope someone could help me with it. Problem is about 2 interacting particles (spin: 1/2 each), with Hamiltonian Ho=-A( S_1z + S_2z) and perturbation H1={(S_1x)*(S_2x) - (S_1y)*(S_2y)}. The question asks to calculate the energies of all 4 states up...

Homework Statement Homework EquationsThe Attempt at a Solution I managed to find dy/dx as follows: But I'm having difficulty finding the second derivative. I've looked at examples using the chain rule but I'm still confused. Would someone mind shedding some light on this for me?

Homework Statement s=0.140406704 2. Relevant equation The Attempt at a Solution So I converted the ODE into the following two equations $$\frac{dx_1}{dt}=x_2$$ $$\frac{dx_2}{dt}=x_1- \alpha sin(x_2)$$ I have done the following with the program so far, I came to a halt because I am not...

Homework Statement Homework Equations Power series The Attempt at a Solution As I have to write in form of "x^2n" & "x^2n+1", I am totally have no idea with how can I go on to do the question. Those I have learned in lecture and online are mostly with only one part of summation... or two...

Homework Statement ##\frac{d^2y}{dx^2}=2xy\frac{dy}{dx}##Homework Equations This is second order non-linear pde of the 'form' ## f(y'',y',y,x) ## . I have read that there are 2 simplified versions of a second order non-linear pde that can be solved easily and these are 1) when there is no y...

Consider a simple single degree-of-freedom (SDOF) spring-mass-dashpot dynamic system with spring rate k, mass m, and viscous damping coefficient c. Dimension x is the absolute displacement of the mass. The base input translation is y. A dot notation indicates differentiation with respect to...

Hello Guys, We haven't yet covered on how to solve 2nd order equation in class however we have this assignment given to us. Any tips would be appreciated for these 2 little problems. 1. Homework Statement We have this initial Equation: d2y/dt2−7dy/dt+ky=0, and we need to find the values of k...

Hi PF! Generally speaking, how would one solve $$f''-a(x) f = 0 : f(0)=0,f'(0)=1$$ Or if you could point me to a source that would be awesome too!

Hi I'm having a slight issue trying to obtain a 2nd order ODE with respect to x (so involves implicit differentiation in this case) from the equation below. I would greatly appreciate any help or tips to solve this problem. I've removed the coefficients to make things a litter easier. Thank you.

Homework Statement Homework EquationsThe Attempt at a Solution For the homogeneous equation, I have got the the root of the characteristic equation as ## e^{ix}, e^{-ix} ## . So, the corresponding solution is ## B \sin{ x} + A \cos{ x} ## . Then, I took the particular solution as...

I got the following two integral for the a particular solution of a 2nd order linear ODE $$(D-a)(D-b)y = g(x)$$ by using inverse operators ##\frac{1}{D-a}## and ##\frac{1}{D-b}##. The two different integrals are obtained by operating these operators in different order on y to get a particular...

I would like to solve the following differential equation, it seems easy but only given one initial value. y''(x) = ln(ln(x)) y(5) = 0 Solve for y(10) I know it can be directly integrated but cannot be expressed in terms of elementary functions. Most numerical method involves expressing the...

Homework Statement Homework EquationsThe Attempt at a Solution Is there anyway to answer this question without solving the eqn and plotting the graph? The function will not oscillate as there is -4y on the right side. So, the first option gets canceled. Since there is a resistive part i.e...

Suppose a spring with spring constant 6N/m is horizontal and has one end attached to the wall and the other end attached to a 3 kg mass. Suppose the friction/damping constant is 1 N s/m Set up a differential equation that describes this system with x denoting displacement of the mass from...

Homework Statement I want to show that $$f''(x) = g(x)$$ has a solution of the form $$f(x) = 2\int_0^{x} dx' (x-x') g(x').$$ It's not hard to verify that it is a solution, the question is how to find it. This should be easy and is likely a standard problem but I haven't found the right...

Homework Statement Write the following second-order ODE as a system of two first-order ODEs. ##d^2y/dt^2 + 5(dy/dt)^2 - 6y + e^{sin(t)} = 0## Homework Equations w = dy/dt The Attempt at a Solution The solution of the book says ##dy/dt = w, dw/dt = -5w - 6y + e^{sin(t)}##, but shouldn't it be...

Hi, I've had obtained a mathematical model for the slip controller issue. As you see I have the diffequation for the slip. and the input that force the system to zero error is provided as well. Now it's time to implement it in simulink or matlab. I took a look at the example provided on...

Hi All, Does anybody know how to solve the following PDE? I tried a similarity solution method where eta = y/f(x) (which I can do successfully without the C * U term) but was unsuccessful. Thank you very much in advance!

Homework Statement Given a set of fundamental solutions {ex*sinx*cosx, ex*cos(2x)} Homework Equations y''+p(x)y'+q(x)=0 det W(y1,y2) =Ce-∫p(x)dx The Attempt at a Solution I took the determinant of the matrix to get e2x[cos(2x)cosxsinx-2sin(2x)sinxcosx-cos(2x)sinxcosx-...