What is 2nd order: Definition and 494 Discussions

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  1. Z

    Difference between resonance in undamped vs damped mass-spring-dashpot

    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...
  2. Z

    Obtaining stability criterion for 2nd order ODEs in coefficient form

    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...
  3. cnh1995

    Engineering What is wrong in this 2nd order transfer function?

    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...
  4. samy4408

    Integrated Rate Law for 2nd Order Reactions

    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 ?
  5. P

    Engineering How do I use Simulink to create a control system with a 2nd order ODE?

    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...
  6. P

    Solving a 2nd order non-linear DE by dimensional analysis/observation

    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'...
  7. D

    Engineering Question - Calculating Coefficients for 2nd Order Transient Analysis

    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...
  8. P

    Solving separable 2nd order DE

    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.
  9. H

    Fourier transform to solve PDE (2nd order)

    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...
  10. greg_rack

    Learning DEs: Solving 2nd Order Differential Equations

    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...
  11. S

    I How to solve 2nd order TDSE for a Gaussian-kicked harmonic oscillator?

    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...
  12. Tony Hau

    How to solve this 2nd order ODE?

    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...
  13. derya

    A Generic Solution of a Coupled System of 2nd Order PDEs

    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...
  14. J

    Help with 2nd order Runge Kutta and series expansion

    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...
  15. Kaguro

    Comp Sci Numerically, how to get the other solution of this 2nd order ODE?

    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?
  16. M

    I Calculating 2nd Order Scattering Amplitude: Feynman Diagrams

    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 ##...
  17. electrogeek

    I 2nd Order Perturbation Theory Energy Correction

    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...
  18. A

    MHB What Method Solves the ODE y''(x) + y'(x) + F(x) = 0?

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

    Angular position for 2nd order diffraction

    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!
  20. Arman777

    Comp Sci Solving 2nd Order DEs with 4th Order RK Method

    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...
  21. JorgeM

    I 2nd order Taylor Series for a function in 3 or more variables?

    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...
  22. karush

    MHB -m30 - 2nd order linear homogeneous ODE solve using Wronskian

    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...
  23. karush

    MHB -a.3.2.96 Convert a 2nd order homogeneous ODE into a system of first order ODEs

    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?
  24. A

    Is this an allowed solution? - 2nd order harmonic oscillation

    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...
  25. A

    Solving 2nd order ODE in order to get equation for Orbital Trajectory

    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...
  26. paulmdrdo

    Engineering Solve 2nd Order RLC Circuit Problem | Get Answers Now

    I was stuck with the last line of my solution please help me find the correct answer to this problem. TIA!
  27. BeyondBelief96

    2nd Order Non-Degenerate TI Perturbation Theory Corrections

    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...
  28. J

    MHB 2nd order differential equation - equation of motion

    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...
  29. Hiero

    I Is there any theoretical basis for laws being 2nd order

    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?
  30. WMDhamnekar

    MHB Solutions of DE System & 2nd Order Differential Equation

    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...
  31. R

    Trying to solve a 2nd order diffy-Q with delta function

    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...
  32. P

    Calculate the energies of all 4 states up to 2nd order

    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...
  33. T

    2nd order differential equations

    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?
  34. CMJ96

    2nd Order Adams-Bashforth/Runge Kutta

    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...
  35. yecko

    2nd order differential equation with power series

    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...
  36. binbagsss

    Solving a Second Order Non-Linear PDE with Undetermined Coefficients

    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...
  37. Tunalover

    I A common 2nd order ODE from dynamics but....

    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...
  38. D

    Solving 2nd order DE with initial condition

    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...
  39. M

    A Solving linear 2nd order IVP non-constant coefficient

    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!
  40. A

    I Trying to obtain a 2nd order ODE

    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.
  41. Pushoam

    Solving 2nd order differential equation

    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...
  42. H

    I Equality of two particular solutions of 2nd order linear ODE

    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...
  43. P

    I 2nd order ODE numerical solution

    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...
  44. Pushoam

    Analyzing 2nd Order Differential Equations with Resistive Components

    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...
  45. ReidMerrill

    2nd order ODE: modeling a spring

    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...
  46. B

    Single-integral solution to 2nd order inhomogeneous ODE

    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...
  47. S

    Write 2nd order ODE as system of two 1st order ODEs

    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...
  48. P

    MATLAB Simulink: designing a 2nd order sliding controller

    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...
  49. K

    A 2nd Order PDE Using Similarity Method

    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!
  50. S

    I Constructing a 2nd order homogenous DE given fundamental solution

    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-...
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