What is First order: Definition and 585 Discussions

In mathematics and other formal sciences, first-order or first order most often means either:

"linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of higher degree", or
"without self-reference", as in first-order logic and other logic uses, where it is contrasted with "allowing some self-reference" (higher-order logic)In detail, it may refer to:

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  1. S

    I Expansion at first order in QCD counterterm

    What is the meaning of the expansion at first order in ##\delta_2## and ##\delta_3## at the second step in the last line? These quantities are not "small" - on the contrary, the entire point is to then take the ##\epsilon \to 0## limit and the counterterms blow up
  2. chwala

    Solve the given first order differential equation

    My thinking is two-fold, firstly, i noted that we can use separation of variables; i.e ##\dfrac{dy}{y}= \sec^2 x dx## on integrating both sides we have; ##\ln y = \tan x + k## ##y=e^{\tan x+k} ## now i got stuck here as we cannot apply the initial condition ##y(\dfrac {π}{4})=-1##...
  3. T

    Solving Coupled First Order ODEs: Is There a Closed Form Solution?

    Hey all, I am currently struggling decoupling (or just solving) a system of coupled ODEs. The general form I wish to solve is: a'(x)=f(x)a(x)+i*g(x)b(x) b'(x)=i*h(x)a(x)+j(x)a(x) where the ' indicates a derivative with respect to x, i is just the imaginary i, and f(x), g(x), h(x), and j(x) are...
  4. dextercioby

    A First order electroweak correction to the g-2 magnetic moment

    We know that we need to go to 5th order in perturbation theory to match 10 decimals of g-2 for electron, theory vs. experiment. But let us not assume QED is pure and independent, but it's a lower energy limit of GSW (not Green-Schwartz-Witten from superstrings) electroweak theory. Has anyone...
  5. B

    Engineering First order differential equations (movement of a rotary solenoid)

    My question i am trying to solve: I have successfully done first order equations before but this one has got me a little stuck. My attempt at the general solution below: $${5} \frac{\text{d}\theta}{\text{d}t}=-6\theta$$ $${5} \frac{\text{d}\theta}{\text{d}t} =\frac{\text{-6}\theta}{5}$$...
  6. M

    Engineering First order non linear to state space equations

    How to represent this system in state space form? where: $$ x' = Ax + Bu \text{ and }y = Cx + Du$$ I am trying to create a state space model based on these equations on simulink, need to find A, B, C and D but like I mentioned, i cannot find the solution when the differentials are not of...
  7. johnsmith7565

    Engineering First Order RL Circuit: Finding a Current I2(0-)

    I shorted the inductor and performed mesh analysis. The solutions to the linear system were done using a calculator. The book says that the value for i2(0-) should be 15 mA but I'm getting -2mA. What am I doing wrong? I'm completely confused. Maybe mesh isn't the most efficient way to find I2...
  8. shivajikobardan

    Comp Sci Mistake while removing implications in first order predicate logic-:

    As you can see I am not getting correct result. What have I messed up? I want to learn it. https://slideplayer.com/slide/4942120/ Here is full slide in case anyone wants to refer to it.
  9. chwala

    Solve the first order differential equation

    From my working...I am getting, ##xy=####\int x^{-1/2}\ dx## ##y##=##\dfrac {2}{x}##+##\dfrac {k}{x}## ##y##=##\dfrac {2}{x}##+##\dfrac {6}{x}## ##y##=##\dfrac {8}{x}## i hope am getting it right...
  10. T

    A Existence of unique solutions to a first order ODE on this interval

    I am trying to find a way to prove that a certain first order ode has a unique solution on the interval (1,infinity). Usually the way to do this is to show that if x' = f(t,x) (derivative with respect to t), then f(t,x) and the partial derivative with respect to f are continuous. However, this...
  11. chwala

    Solve this first order differential equation

    This is the question; This is the solution; Find my approach here, ##x####\frac {dy}{dx}##=##1-y^2## →##\frac {dx}{x}##=##\frac {dy}{1-y^2}## I let ##u=1-y^2## → ##du=-2ydy##, therefore; ##\int ####\frac {dx}{x}##=##\int ####\frac {du}{-2yu}##, we know that ##y##=##\sqrt {1-u}## ##\int...
  12. D

    I Usage of First Order Elastic Constants in Soft Body Equations

    Hi, I have some soft body equations that require first order elasticity constants. Just trying to figure out the proper indexing. From Finite Elements of Nonlinear Continua by J.T. Oden, the elastic constants I am trying to obtain are the first order, circled below: My particular constitutive...
  13. B

    MHB First order differential equations

    Hi, Is the answer: y(x) _homogenous =v(x) y(x) _private =u(x)v(x) ? Or they refer to something else? I don't know how to approach to it
  14. A

    Solving a first order differential equation with initial conditions

    Hello! Consider this ODE; $$ x' = sin(t) (x+2) $$ with initial conditions x(0) = 1; Now I've solved it and according to wolfram alpha it is correct (I got the homogenous and the particular solution) $$ x = c * e^{-cos(t)} -2 $$ and now I wanted to plug in the initial conditions and this is...
  15. DaveC426913

    B First order approx. for a curve

    Helping someone with some fictional physics. He's looking for a function that will produce a curve similar to this (poor geometry is my doing, assume smooth curvature): Starts at 0,0. Maximum at n. Reaches zero at infinity. The cusp is not sharp, it's a curve (which, I think suggests at least...
  16. alan123hk

    I How do I solve this first order second degree differential equation?

    How to solve this first order second degree differential equation ? ##\left(\frac {dy} {dx}\right)^2 + 2x^3 \frac {dy} {dx} - 4x^2y=0 ## Thanks.
  17. Lilian Sa

    First order differential equation involving a square root

    Summary:: solution of first order derivatives we had in the class a first order derivative equation: ##\frac{dR(t)}{dt}=-\sqrt{\frac{2GM(R)}{R}}## in which R dependent of time. and I don't understand why the solution to this equation is...
  18. V

    First Order Logic: ∀x,y a(x) ∧ a(y)

    An example we were given is as follows: {ua|u∈∑*} (where ∑* is set of all words over ∑) so we have ∀x. last(x) → a(x). I am given {awa|w∈∑*} to do, and I know that I have to express that a is the first letter and last letter in a word. Could I write it as: ∀x,y ( a(x) ∧ a(y) ∧ x<y → ∃z(x<z<y))...
  19. B

    Solving 1D First Order Equations for 3D Mass Positions and Velocities

    Okay so I need to find 12 one dimensional first order equations that describe the position and velocity of both masses in 3 dimensions. The equations for the second body will be easy once I figure out how to do the first body, so I'll ignore that for now. For the first equation, I can rearrange...
  20. L

    A Non linear diff. eq. first order

    I tried to solve for y' the algebric second grade equation but I don't know how to solve the diff. equation either. Any help? Thanks. -- lightarrow
  21. Mayhem

    I Using v substitution for first order homogenous DE and constraining solution

    My considers a type of differential equation $$\frac{\mathrm{d} y}{\mathrm{d} x} = f\left(\frac{y}{x} \right )$$ and proposes that it can be solved by letting ##v(x) = \frac{y}{x}## which is equivalent to ##y = xv(x)##. Then it says $$\frac{\mathrm{d} y}{\mathrm{d} x} = v + x\frac{\mathrm{d}...
  22. T

    First Order Diffy Q Problem with Bernoulli/Integrating Factors

    I seem to be getting an unsolvable integral here (integral calculator says it's an Ei function, which I've never seen). My thought was to use Bernoulli to make it linear and then integrating factors. Is that wrong? The basic idea is below: P(x) 1, Q(x) = 1/2(1-1/x), n=-1, so use v=y^1-...
  23. Julio1

    MHB Solutions of the ODEs - 2 first order linear equations

    Find the general solution of the ODE: $\check{X_1}=X_1$ $\check{X_2}=aX_2$ where $a$ is a constant.
  24. karush

    MHB -b.2.2.33 - Homogeneous first order ODEs, direction fields and integral curves

    $\dfrac{dy}{dx}=\dfrac{4y-3x}{2x-y}$ OK I assume u subst so we can separate $$\dfrac{dy}{dx}= \dfrac{y/x-3}{2-y/x} $$
  25. karush

    MHB -b.2.2.32 First order homogeneous ODE

    \[ \dfrac{dy}{dx} =\dfrac{x^2+3y^2}{2xy} =\dfrac{x^2}{2xy}+\dfrac{3y^2}{2xy} =\dfrac{x}{2y}+\dfrac{3y}{2x}\] ok not sure if this is the best first steip,,,, if so then do a $u=\dfrac{x}{y}$ ?
  26. karush

    MHB -2.2.31 First order homogeneous ODE

    I OK going to do #31 if others new OPs I went over the examples but? well we can't 6seem to start by a simple separation I think direction fields can be derived with desmos
  27. P

    A First order formalism of Polyakov action

    In the notes of Arutyunov, he writes down the equation of Polyakov action in what he calls a first-order formalism(equation 3.19). But here I did not understand how he got this equation. Can someone help? Moreover, can someone explain how he got the constraints in equation 3.25? And why they...
  28. karush

    MHB -2.2.27 - Analysis of first order IVP

    well each one is a little different so,,, $$\dfrac{dy}{dt}=\dfrac{ty(4-y)}{3},\qquad y(0) =y_0$$ not sure if this is what they meant on the given expression
  29. karush

    MHB -b.2.2.26 Solve first order IVP and determine where minimum of solution occurs

    OK going to comtinue with these till I have more confidence with it $$\dfrac{dy}{dx}=2 (1+x) (1+y^2), \qquad y(0)=0$$ separate $$(1+y^2)\, dy=(2+2x)\, dx$$
  30. P

    MHB Proving First Order Logic in Machover's Text

    Trouble working through Set theory, Logic, and their Limitations by Maurice Machover. Particularly these 1. $\sigma \vDash \alpha \rightarrow \forall x\alpha$ where $x$ does not occur in a free $\alpha$ 2. $\sigma \vDash s_1 = t_1 \rightarrow ... \rightarrow s_n = t_n \rightarrow...
  31. H

    B Question about expanding a function to first order

    If we have a function ##f(x+\Delta x)## where ##\Delta x << x##, is it valid to approximate this as: $$f(x + \Delta x) \approx f(x) + f'(x)\Delta x$$ even if ##\Delta x## is not necessarily small? If not, what is the valid expansion to first order?
  32. karush

    MHB -7.1 transform u''+0.5u'+2u=0 into a system of first order eq

    transform the given equation into a system of first order equation$$u''+0.5u'+2u=0$$ok from examples it looks all we do is get rid of some of the primes and this is done by substitutionso if $u_1=u$ and $u_2=u'_1$ then $u_2=u'$ and $u'_2=u''$ then we have $u'_2+0.5u_2 +2u_1 = 0$then isolate...
  33. 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?
  34. J

    A First order logic and set theory: who comes first?

    Goldrei's Propositional and Predicate Calculus states, in page 13: "The countable union of countable sets is countable (...) This result is needed to prove our major result, the completeness theorem in Chapter 5. It depends on a principle called the axiom of choice." In other words: the most...
  35. Haorong Wu

    I What is "to the first order in H"?

    I'm learning Griffiths' QM (3rd edn). In Chapter 11 (Quantum Dynamics), there is an expression I'm not familiar with: ## \left| C_b \right|^2 = \left[ - \frac i \hbar \int_0^t H'_{ba} \left( t' \right) e^{i \omega_0 t'} \, dt' \right] \left[ \frac i \hbar \int_0^t H'_{ba} \left( t'...
  36. S

    Regression for a first order system

    Homework Statement I am carrying out a regression for diameter of a part Homework Equations Diameter = -0.0531052 + 0.0443237 * exp (-0.0103633 * 'Time elapsed') if diameter is -0.052 then can some one please calculate the value for time elapsed would you please explain the steps The...
  37. J

    I Solution for 1st order, homogenous PDE

    ##u_t + t \cdot u_x = 0## The equation can be written as ##<1, t, 0> \cdot <d_t, d_x, -1>## where the second vector represents the perpendicular vector to the surface and since the dot product is zero, the first vector must necessarily represent the tangent to the surface. We parameterize this...
  38. chwala

    Solving a first order ODE using the Adomian Decomposition method

    Homework Statement how do we solve the ode ## y'+y^2=-2, y(0)=0## using adomian decomposition method?Homework EquationsThe Attempt at a Solution ##Ly = -2-y^2## ## y= 0 + L^{-1}[-2-y^2]## ##y_{0}= -2t## ##y_{1}= -L^{-1}[4t^2] = -4t^3/3## are my steps correct so far in trying to get the Adomian...
  39. J

    Comp Sci Help with solving first order ODE using a simple Fortran code, please

    I am trying to solve the following first order ODE using a simple Fortran code : $$ ds/dt=k_i * \sqrt{v}$$ where both (ki) and (v) are variables depending on (h) as follows $$ k_i=\sqrt{χ/h^2}$$ $$v= \mu h$$ where (μ) and (χ) are constants. (the arbitrary values of each of them can be seen...
  40. GregBrown

    Exploring Euler's Formula in First Order Chemical and Nuclear Reaction Kinetics

    Has anyone ever encountered a discussion on the topic of applying Euler's formula exp(i*x) = cos(x) + i * sin(x) to the equation governing first order chemical (and nuclear) reaction kinetics? d[Reactant]/dt = C*[Reactant]
  41. WMDhamnekar

    MHB Difficult first order linear differential equation

    Hello, I want to solve the following differential equation. $y'=\dfrac{x^3-y^3}{x-y}$. How to solve it?
  42. Chromatic_Universe

    I Solving a nonlinear first order differential equation

    (a'[t]/a[t])^2 == K*(A + B*a[t]^-6)^1/2} is the equation to be solved for getting the solution of a(t) in terms of time(t). Any ideas on how to solve this problem? Use of Matlab or Mathematica is accepted.
  43. Felipe Lincoln

    First order differential equation

    Homework Statement Solve the following differential equation such that ##x(0)=1##. ## \dfrac{dx}{dt} + 2tx = 3e^{-t^2}+t## Homework Equations Integrating factor: ##\mu(t) = exp\left(\int_0^t2t \right)## The Attempt at a Solution I used the integrating factor and then got the solution ##x(t) =...
  44. Buckethead

    B What is meant by "first order" and "second order"

    I see comments such as "explains ... to the first order" or "to the second order" quite a bit in physics discussions. Can someone explain in lay terms, what first order and second order refer to?
  45. E

    First order perturbation energy correction to H-like atom

    Homework Statement Real atomic nuclei are not point charges, but can be approximated as a spherical distribution with radius ##R##, giving the potential $$ \phi(r) = \begin{cases} \frac{Ze}{R}(\frac{3}{2}-\frac{1}{2}\frac{r^2}{R^2}) &\quad r<R\\ \frac{Ze}{r} &\quad r>R \\...
  46. L

    I Second order ordinary differential equation to a system of first order

    I tried to convert the second order ordinary differential equation to a system of first order differential equations and to write it in a matrix form. I took it from the book by LM Hocking on (Optimal control). What did I do wrong in this attachment because mine differs from the book?. I've...
  47. L

    Convert to a first order ODE

    What am I doing wrong here in my attachment?
  48. renec112

    First order pertubation of L_y operator

    Hi, I am trying to solve an exam question i failed. It's abput pertubation of hydrogen. I am given the following information: The matrix representation of L_y is given by: L_y = \frac{i \hbar}{\sqrt{2}} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1...
  49. Mr Davis 97

    Interpreting a statement in first order logic

    Homework Statement Rewrite the following statements in symbolic form: a) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a solution. b) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a unique solution. Homework EquationsThe Attempt at a...
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