Second order Definition and 602 Threads

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence




P


x
(
P
x

¬
P
x
)


{\displaystyle \forall P\,\forall x(Px\lor \neg Px)}
says that for every formula P, and every individual x, either Px is true or not(Px) is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables as explained in the section Syntax and fragments. Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified.

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  1. thegirl

    Why is the wave equation a second order differential?

    I don't know if this is a silly question? Am I missing simple math? How does a wave depending on amplitude and frequency make it's equation a second order differential equation?
  2. A

    Solve second order nonlinear differential equation

    how do you solve this equation? y´´ + k/(y^2) = 0 ? I got it from applying Newton's 2nd law of motion to an object falling from space to Earth only affected by gravitational force. Thank you!
  3. VoteSaxon

    Finding the Particular Integral for d2y/dt2 + 4y = 5sin2t

    Homework Statement Well I am looking for the particular integral of: d2y/dt2 + 4y = 5sin2t The attempt at a solution As f(t) = 5sin2t, the particular integral yPI should look like: yPI = Acos2t + Bsin2t dyPI/dt = -2Asin2t + 2Bcos2t d2yPI/dt2 = -4Acos2t - 4Bsin2t Subbing into the differential...
  4. V

    Typical examples of second order nonlinear dynamic systems

    Hi guys, after hours of searching internet I couldn't find much real-life examples of second order nonlinear dynamic systems (only tons of tons of equation and system theory... got totally frustrated). They will serve as a base process for modeling controllers. So far I found propeller pendulum...
  5. kostoglotov

    Question concerning 2nd order homogeneous linear diff eqs

    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...
  6. B

    How to reduce a system of second order ODEs to four first order equations?

    Someone can explain me how to get the general solution for this system of ODE of second order with constant coeficients: \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix} \begin{bmatrix} \frac{d^2x}{dt^2}\\ \frac{d^2y}{dt^2}\\ \end{bmatrix} + \begin{bmatrix} b_{11} &...
  7. Remixex

    Hooke's and Newton's law to find Second order ODE

    Homework Statement A weight of 8 pounds extends a spring 2 feet. It's assumed that the damping force that acts on the system is equal (number-wise) to alpha times the speed of the weight. Determine the value of alpha > zero so x(t) is critically damped. Determine x(t) if the weight is liberated...
  8. can12345

    First and second order terms in pressure measurement

    Hello everyone, What does it means first and second order of pressure measurement value? Whats the main difference?
  9. M

    How to Derive Differential Equations for a Second Order System?

    I have attached an image of a question I am trying to do, I want to find the differential equations that describe the second order system in the image. I know for a spring, potential energy = 1/2.K.x (where k is the spring constant, and x is the distance the spring is stretched). I know that...
  10. M

    MHB Taylor expansion of second order

    Hey! :o I have to find the Taylor expansion of second order of the following functions with center the given point $(x_0, y_0)$. $f(x, y)=(x+y)^2, x_0=0, y_0=0$ $f(x, y)=e^{-x^2-y^2}\cos (xy), x_0=0, y_0=0$ I have done the following: The Taylor expansion of second order of $f...
  11. I

    Laplace Transform of A Second Order ODE

    23.) y'' + 2y' + y = 4e-t; y(0) = 2, y'(0) = -1 Y(s) = [(as + b) y(0) + a y'(0) + F(s)]/(as2 + bs + c) My attempt: a = 1, b = 2, c = 1 F(s) = 4 L{ e-t } = 4/(s+1) (From Laplace Transform Table) Plugging and simplifying: Y(s) = (2s2 + 5s + 7)/[(s + 1)(s2 + 2s + 1) Here is where I get...
  12. nomadreid

    Second order curiosities: MPP, implication

    In setting up a syntax for a first order theory, one usually includes Modus Ponens as a metarule. However, couldn't MP just be rewritten as a second-order sentence, thereby making all supposedly first-order theories de facto second-order ones? While on the subject of second order theories, I...
  13. Feodalherren

    Y′′=−20⋅4x^3, Second order linear ordinary DE

    Homework Statement y′′=−20⋅4x^3 Homework Equations Undetermined coefficients method The Attempt at a Solution so at first, solving the associated homogeneous equation I find the fundamental set of solutions to be: y1=1 and y2=x. I know that these are correct. Now for the part that confuses...
  14. manifold

    A numerical solution of a second order ODE

    Hello everyone; i'd like some help in this problem : i want to solve num this differential equation { y"(t)+t*cos(y)=y } by the Taylor method second order expansion. i first have to make this a first order differential equation by taking this vector Z=[y' y] then we have Z'=[y" y'] which equal...
  15. powerof

    Symmetry in second order partial derivatives and chain rule

    When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...
  16. Alexandre

    Is this correct second order approximation?

    I have a second order differential equation of the form (theta is a function of time): \theta ''=F\left(\theta ,\theta '\right) Turning them to two first order equations I get: \begin{cases} \theta '\:=\omega \\ \omega '=F\left(\theta ,\omega \right) \end{cases} And here's the algorithm...
  17. F

    MHB Finding interval where second order ODE has unique solution

    I'm a little stuck getting started on this question. y''+\tan(x)y=e^x with y(0)=1,y'(0)=0. I know the existence and uniqueness theorem for an nth order initial value problem How do I apply the theorem?
  18. L

    Second order differential equation

    Homework Statement So I'm in pchem right now and I haven't taken dif eq (it's not required, but I wish I had taken it now!) I am asked to solve this differential equation: y''+y'-2y=0 Homework Equations I know for a second order differential equation I can solve for the roots first. If...
  19. S

    Wave second order derivative equation

    Whenever the second order derivative of any physical quantity is related to its second order space derivative a wave of some sort must travel in a medium, why this is so?
  20. S

    General solution of second order ODE

    Homework Statement Find the general solution. Homework Equations y"+y=x2sin2x The Attempt at a Solution Characteristic equation would be: m2 + 1 = 0 So,m2 = -1 Therefore, m = i or m = -i. Complementary function would be : Asinx+Bcosx where,A and B are constants respectively. If I write...
  21. C

    System of two second order ODE's

    Hi , I have tried solving the following system of ODE's (eq1 attached) using Matlab. first i reduced it into a system of four 1st order ODE's (eq2 attached). thani tried to solve it using ode45() in the following manner: function xprime = Mirage(t,x); k=2; xprime=[x(1)...
  22. T

    Understanding the Second Order Relation of Entropy: A Homework Guide

    Homework Statement Find: Homework Equations The Attempt at a Solution
  23. B

    Second Order Derivative Notation (mingled with)

    I've been thinking about something recently: The notation d2x/d2y actually represents something as long as x and y are both functions of some third variable, say u. Then you can take the second derivatives of both with respect to u and evaluate d2x/du2 × 1/(d2y/du2). Now I think it's also...
  24. mr_sparxx

    Very simple: second order derivative in wave equation

    In the equation regarding an array of masses connected by springs in wikipedia the step from $$\frac {u(x+2h,t)-2u(x+h,t)+u(x,t)} { h^2}$$ To $$\frac {\partial ^2 u(x,t)}{\partial x^2}$$ By making ##h \to 0## is making me wonder how is it rigorously demonstrated. I mean: $$\frac {\partial ^2...
  25. icesalmon

    Second order ODE for RLC circuit

    if I consider a circuit consisting of a capacitor, an inductor and a resistor and using kirchhoffs voltage rule for the circuit i come up with the following L(Q''(t)) + R(Q'(t)) + (Q(t))/C = 0 I solve for the roots using a characteristic equation of the form LM2 +MR +(1/C) = 0 solving this for...
  26. C

    How Can We Solve this Second Order ODE for Electron Behavior?

    Homework Statement I'm taking an online introductory chem course, and while explaing the failure of classical mechanics to describe electron behavior, the teacher brought up the following ode which is based on Newton's second law and coulombs law: -e^2/4(pi)(epsilon-nuaght)r^2=m(d^2r/dt^2)...
  27. 2

    System of two second order ODE's. Solution does not agree with Wolfram.

    Homework Statement Solve the following system of differential equations: ##y''(x) = y'(x) + z'(x) - z(x)## ##z''(x) = -5*y'(x) - z'(x) -4*y(x) + z(x)## 2. The attempt at a solution I converted the two second order equations to 4 first order equations by substituting: ##g(x) = y'(x)## and...
  28. C

    Oddly Formatted Second Order ODE

    Homework Statement u'' + w20*u = cos(wt) w refers to omega. Homework EquationsThe Attempt at a Solution I'm not sure where to begin on this. For starters, it's a multiple choice problem, and all the answers are given in terms of y, so I'm not sure if u is supposed to replace y' or something...
  29. _N3WTON_

    Second Order ODE, Complex Roots, Change of Variables

    Homework Statement Solve: \frac{d^{2}y}{dx^{2}} + \omega^{2}y = 0 Show that the general solution can be written in the form: y(x) = A\sin(\omega x + \alpha) Where A and alpha are arbitrary constants Homework EquationsThe Attempt at a Solution I know that I will need to change variables for...
  30. C

    Second Order ODE, With Initial Conditions

    Homework Statement y'' + 4y = t2 + 6et; y(0) = 0; y'(0) = 5 Homework Equations The Attempt at a Solution So, getting the general solution, we have r2 + 4 = 0, so r = +/- 2i So the general solution is yc = sin(2t) + cos(2t) I then used the method of undetermined coefficients to figure that...
  31. V

    Engineering Solving a Second Order Circuit for Capacitor Voltage

    Homework Statement Homework Equations Here is the technique I am using: The Attempt at a Solution [/B] I understand how to solve the problem using the technique provided by the solution but I was wondering where I messed up in the technique that I used. I prefer the second...
  32. Y

    Euler's Method to approximate a second order Differential Equation

    Homework Statement y'' + 4y' + 4y = 0 ---- y(0) = 1, y'(0) = 5 Find the exact solution of the differential equation. Use the exact solution and Euler's Method to compute Euler's Approximation for time t = 0 to t = 5 using a step h=0.05. Plot Euler's & Exact vs. t and plot Error vs. t. Then...
  33. 2

    Second order nonlinear ODE. How to begin solving it?

    Homework Statement This is not the exact problem that I want to solve but I will use this as a guidance tool: ##y'' - (y')^2 + y^3 = 0## where y is the function of x 2. The attempt at a solution I tried doing a substitution ##u(x) = y'(x)## which leads to ##u' - u^2 + y^3 = 0## where both u...
  34. D

    MHB How Can I Solve This Second Order Linear ODE Problem?

    I'm having a lot of trouble with this problem. I'm also having a lot of trouble inputting it into LaTeX. I hope you can follow even though the markup isn't good. I'm trying to find a formula for the general solution of $ax^2y''+bxy'+cy=0$ where $y=x^r\ln(x)$ when $(b-a)^2-4ac=0$; using...
  35. DivergentSpectrum

    How do I apply rk4 to a second order pde?

    Im writing a program that calculates the trajectory of a particle in an arbitrary force field. the force field is a vector function of position (x, y, z) AND velocity (x', y', z'). Rk4= runge kutta forth order method Please help. Thanks!
  36. DivergentSpectrum

    Numerical second order pde solver

    Edit:whoops wrong forum mods please move 2nd edit: I just had dinner then got back on the computer, input some points and saw a beautiful elipse.(complete with a fascinating flower petal design due to inaccuracies) Weird lol! No idea why it wasnt working before Now to implement RK4 bwahahaha...
  37. J

    Euler-lagrange, positivity of second order term

    For twice differentiable path x:[t_A,t_B]\to\mathbb{R}^N the action is defined as S(x) = \int\limits_{t_A}^{t_B} L\big(t,x(t),\dot{x}(t)\big) dt For a small real parameter \delta and some path \eta:[t_A,t_B]\to\mathbb{R}^N such that \eta(t_A)=0 and \eta(t_B)=0 the action for...
  38. F

    Complex contour integral with a second order pole at origin

    Homework Statement Hello all. I'm currently attempting to prove the central limit theorem using a simple case of two uniformly distributed random variables. Aside from being able to solve it using convolutions, I also wish to solve it by using the Dirac Delta function. That aside, the integral...
  39. M

    A question on second order linear equations

    Hi, all. While solving a second order linear differential equation why do we have to use linear independent but two solutions. For example, when solving y''- y = 0 , y(0) = 5 and y'(0) = 3 , we use ex and e-x and then we write y = c1*ex+ c2*e-x-
  40. S

    Solutions of second order linear PDEs

    Question about Solutions of second order linear PDEs I don't have very much formal knowledge of this topic, this is something I have been thinking about, so excuse me if my notation is off. I have a question about second order linear PDEs, do all have a separable solution? It seems that we can...
  41. K

    Understanding Bode Plots for Second Order Systems with ς‚= 0

    hey guys, i have a question regarding bode plot g(s)= 1/(s2+4) i did get the magnitude plot correct but i am unable to understand the phase plot. by calculating on paper i got 0° but in MATLAB it changes from -360° to -180° i haven't understood how the initial phase is -360 which...
  42. M

    Second order PDE (w.r.t 2 variables)

    Homework Statement find the solution to: \frac{\partial^{2}u}{\partial x \partial y} = 0 \frac{\partial^{2}u}{\partial x^{2}} = 0 \frac{\partial^{2}u}{\partial y^{2}} = 0 Homework Equations theorem of integration The Attempt at a Solution now from a previous question I...
  43. V

    How to integrate second order derivative

    I would like to know how do we solve d2x/dt2 = k' where k' is a constant i.e the task is to find x as a function of time ? One way to approach this is to rewrite it as vdv/dx = k' where v=dx/dt and first find find v as a function of x and then rewrite v as dx/dt and then find x as a function...
  44. nomadreid

    Is Gödel numbering a first or second order function?

    Gödel-numbering (in its broadest meaning, not necessarily the one Gödel used): On one side, it would seem that an assignment of a symbol to a number is just a first order function, and the recursion set up to translate a formula into numbers would be first-order, but on the other hand the...
  45. N

    Second order differential equation form

    A second order differential equation form d2y/dx2 = f(x,y,dx/dy) How do I read the language on the right hand side?
  46. M

    Truncation Error and Second Order Accuracy.

    By combining the two equations i should be able to solve for u' and get rid of u'': u_(i-1) = u_i + (-h)*u' + 1/2 * (-h)^2 * u'' + O(h^4) u_(i-2) = u_i + (-2h)*u' + 1/2 * (-2h)^2 * u'' + O(h^4) But i keep getting stuck and can't come up with the answer below. Can anyone help me please...
  47. N

    Symbolic solve coupled second order differential equations

    Dear all, I have posted a similar question in another forum and the general consensus seems to suggest that it is not possible to symbolic solve a system of coupled second order different equation with damping (dissipation) and driving forces. However, I have found in many papers and books...
  48. C

    Second order system of odes with variable coefficients

    Hi, I have looked everywhere. Can someone please point me in the right direction for solving a system of ODEs with variable coefficients? I managed to solve such system with constant coefficients.
  49. MarkFL

    MHB Solve Linear Inhomogeneous 2nd Order ODE - Alvin's Question on Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  50. Ravi Mohan

    Solving second order coupled differential equation

    How do we solve a system of coupled differential equations written below? -\frac{d^2}{dr^2}\left( \begin{array}{c} \phi_{l,bg}(r) \\ \phi_{l,c}(r) \\ \end{array} \right)+ \left( \begin{array}{cc} f(r) & \alpha_1 \\ \alpha_2 & g(r)\\ \end{array} \right).\left( \begin{array}{c}...
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