What is Second order: Definition and 602 Discussions

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. R

    On the regular singularities of a second order differential equation

    Homework Statement The only singularities of the differential equation y''+p(x)y'+q(x)y=0 are regular singularities at x=1 of exponents \alpha and \alpha', and at x=-1 of exponents \beta and \beta', the point at infinity being an ordinary point. Prove that \beta=-\alpha and \beta'=-\alpha'...
  2. L

    Compute the second order Taylor polynomial centered at 2 for ln(x)

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  3. G

    Differential Equations - Second Order

    Homework Statement Hi, It's been a while since I have taken differential equations. How do I solve an equation like this: k_{1}\frac{d^{2}V_{x}(t)}{dt^{2}}+k_{2}\frac{dV_{x}(t)}{dt}+k_{3}V_{x}(t)=0 Homework Equations The Attempt at a Solution I have looked through my...
  4. A

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  5. M

    Under-damped second order system - CONTROL

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  6. A

    A system of 200 second order differential equations

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  7. S

    RLC Circuit Second Order Differential Equation

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  8. B

    Engineering Second order circuit need some confirmation on the steps involved.

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  9. Ryuzaki

    Help with Annihilator Method for Non-homogeneous Second Order D.Es

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  10. M

    Second Order Equation - Change of variables

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  11. J

    A second order nonlinear ode in an electrostatics problem

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  12. S

    Second Order Differential Equation (homogenous)

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  13. P

    Finding Constants in Second Order DE with Given Conditions

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  14. B

    How to solve a second order diff eq?

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  15. S

    Thoughts on a non-linear second order problem

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  16. 3

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  17. E

    Can Fourier Series Simplify Solving Nonlinear ODEs with Oscillatory Inputs?

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  18. A

    Sum of Second Order Linear PDEs

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  19. M

    First and Second Order Systems - Classical Analysis

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  20. S

    Second order pde - on invariant?

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  21. M

    Numerically Solving a Second Order Differential Equation Using C

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  22. R

    Linearization of Second Order Differential Equations

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  23. D

    How Do You Solve a Second Order ODE with Specific Boundary Conditions?

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  24. J

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  25. I

    Second order expansion of metric in free-fall

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  26. S

    Mathematica How to Solve a Second Order PDE in Mathematica?

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  27. H

    Solving ODE's or Euler second order diff. eq's containing Asecx?

    I know how we solve ODE's and euler equations in which we have cos and/or sin terms on the right. We take the particular solution to be Acos(x) + Bsin(x). But what if we have secant or cosecant terms on the right or tan and/or cotangent terms? Qno. 1 Are these 4 terms possible i.e. can they...
  28. C

    Second order DE quick question

    I don't understand where the A and B come from, if y = e^(mx), would the general solutions be y = Ae^(mx) + Be^(m_1x) assuming there are two distinct roots of the auxiliary equation? If anyone could clear this up, thanks.
  29. fluidistic

    Variation of parameters for a second order ODE

    Homework Statement I must solve ##y''+2y'+2y=e^{-t}\sin t##. I know variation of parameters might not be the fastest/better way to solve this problem but I wanted to practice it as I never, ever, could solve a DE with it. (Still can't with this one). Though the method is supposed to work...
  30. fluidistic

    Second order ODE, I think 2 regular points

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  31. C

    Second order ODE with RHS = product of two functions

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  32. S

    Simple Pendulum nonlinear second order differential equation

    Homework Statement given: dt=-\frac{1}{2}\sqrt{\frac{l}{g}}\frac{d\theta}{\sqrt{sin^2(\alpha/2)-sin^2(\theta/2)}} make the change of variables sin(\theta/2)=sin(\alpha/2)sin(\phi) to show that: dt=-\sqrt{\frac{l}{g}}\frac{d\phi}{\sqrt{1-k^2sin^2(\phi)}} where k=sin(\alpha/2) Homework...
  33. C

    Conceptual Second order differential eqn question

    I know that if ##Y_1## and ##Y_2## are two solutions of a nonhomogeneous second order differential eqn, then ##Y_1 - Y_2## is also a solution. So this motivates the following: if we set ##Y_1 = y(x)##, where ##y(x) ## is an arbritary soln of the nonhomogeneous ODE and ##Y_2 = y_p(x)##, some...
  34. J

    How do I find di_L(0)/dt for a second order RLC circuit after the switch opens?

    http://imgur.com/6aAMV So I need to find the current labeled as a function of time. THe switch opens at t=0 and I drew the circuit after the switch opens. I found the initial current to be -4 and the voltage on the capacitor to be 8. I'm having trouble trying to find di_L(0)/dt. I...
  35. Z

    Standard Form for second order systems.

    Suppose there's a system with given uncertain parameters. And I would like to obtain certain Rise time, max. over shoot, settling time by adjusting those parameters. Let's say this is the second order system; T(s) = (ks + c) / (s2 + as + b) First of all; for a second order system there...
  36. M

    Nonlinear second order differential equation

    I am having a problem finding the solution for this eq: y''(x)+(2/x)y'(x)+(w^2)y(x)=0 I couldn't find examples in the textbook that goes on a similar line, and have been searching the internet as well, but no use. I am thinking of using substitution v=y' but not sure how to do that in the...
  37. T

    Nonlinear second order ODE describing a force field

    Not sure if this topic belongs here, but here goes. Homework Statement From the AP physics C 1995 test there is a problem that gives the potential energy curve U(x). With F=-\frac{dU}{dx} in one variable, F(x)=-\frac{a}{b}+\frac{ba}{x^{2}} Where a and b are constants. Now I need to get...
  38. fluidistic

    Solving a second order DE via Green's function

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  39. X

    Power Series to solve Second order Differential Equations

    Homework Statement When solving a D.E. with power series, I've encountered something along the lines of: (2 - r)^{2}g'' = -2 Homework Equations Power Series The Attempt at a Solution I know I am just supposed to assume such a series exists, and work from there. But I'm really...
  40. O

    Drawing the step response of a second order system

    Homework Statement I have to draw the step response of the following two systems. G1 = (4+3s)/(s^2+4s+4) G2 = 3/(s^2+4s+4) So I started to draw the step response of the second system first. It has to be in the funky standard form: \frac{ω2}{s2 + 2ζωs + ω2} EDIT: Seems like the above doesn't...
  41. chisigma

    MHB Another second order non homogeneous ODE....

    Four days ago on mathhelpforum.com the user ssh [I don’t know if he the same as in MHB…] has proposed the following second order complete linear ODE… $\displaystyle y^{\ ''} – \frac{2+x}{x}\ y^{\ ’}\ + \frac{2+x}{x^{2}}\ y = x\ e^{x}$ (1) … and till now no satisfactory solution has been...
  42. N

    Particular Solution of 2nd Order ODE: x^2y"-4xy'+6y=ln(x)

    I was wondering what a guess would be for the particular solution of the right hand side of an equation if it looked like this: x^{2}y" - 4xy' + 6y = ln(x) My textbook has some specific examples of the right side function along with the corresponding form of the particular solution...
  43. D

    Removing a Singularity for a Second Order ODE

    I was given the following equation to solve: x^2*y'' + x*y' + k^2*x^2*y = 0 B.C. y'(0)=0, y(1)=0 where k is just some constant. I am having a hard time removing the singularity created by the boundary condition at y' and not aware of a method how. Any advice would be greatly appreciated.
  44. S

    MHB Solving Second order non - homogeneous Differential Equation

    How to solve \( (x+1) y'' - (2x+5) y' + 2y = (x+1) e^x\) can we assume \(y_1 = (Ax+B) e^x \), then \(y_2= vy_1​\) Is this right? then solve for A and B Finally \( y = c_1 y_1 + c_2 y_2\)
  45. STEMucator

    Proof of Second Order ODE Theorem

    Homework Statement I'm pretty sure this is a typo? http://gyazo.com/802746486cc68852e5384d5a12aed596 Homework Equations See the image ^. The Attempt at a Solution I believe the theorem they're talking about, is that you can write the general solution of a second order ODE : L[y] = y'' +...
  46. T

    Differentials/ second order/ electrical vibration

    Homework Statement A simple series circuit has an inductor of 1 henry, a capacitor of 10^-6 farads, and a resistor of 1000 ohms. The initial charge on the capacitor is zero. If a 12V battery is connected is connected to the circuit, and the circuit is closed at t=0, find the charge on the...
  47. T

    Springs Question - Second Order O.D.E

    Hi all, I was wondering is you could help me with this springs question. We've only done springs hanging from a fixed support above being stretched, but now I've got a question where the spirng is being compressed. Homework Statement So, here's some basic info about the question...
  48. C

    A Second Order Differential Equation

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  49. C

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  50. D

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