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. J

    Solving Nonexact First Order ODEs.

    Homework Statement Solve (x - \sqrt{xy})dy - ydx = 0 Rearranged gives us -y + (x - \sqrt{xy})y' = 0 And it looks like an exact differential equation, but is it really? Homework Equations For any given exact equation of the form M(x,y) + N(x,y)y' = 0 The following must be true...
  2. T

    First order, non linear ode problem

    Homework Statement Finding the general solution of y'= (2y^2)/e^x and the particular given that y(0) = 0 The Attempt at a Solution I separate variables etc. to get y = -1/(2c-2e^(-x)) where c is the constant. The problem then is obtaining the particular solution. If I substitute y and...
  3. W

    Laplace Transforms to solve First Order ODE's.

    Homework Statement Solve the system of first order differential equations using Laplace Transforms: dx/dt = x - 4y dy/dt = x + y, subject to the initial conditions x(0)=3 and y(0)=-4. Homework Equations So far I've used the limited knowledge of Laplace Transforms for first order...
  4. K

    Solving First Order Linear Inhomogenous Eq.

    y'= \frac{y}{1+e^x}+e^{-x} It's an easy first order linear inhomogenous eq. I solved it by hand with the formula that one can find anywhere AND with Mathematica, but when I take the derivative to check the solution it comes out wrong and it's freaking me out. Can anyone here post just the...
  5. J

    First order phase transition (van der waals gas)

    I am looking at some of the notes but don't quite understand this. What are the physical explanation of the graphs (Fig 4(a) and 4(b)) on Page 4 ? http://www.pma.caltech.edu/~mcc/Ph127/b/Lecture3.pdf" Why V_{g} decreases with temperature but V_{l} increases with temperature?
  6. J

    First order phase transition (van der waals gas)

    Homework Statement Give the physical explanation of the graphs (Fig 4(a) and 4(b)) on Page 4 http://www.pma.caltech.edu/~mcc/Ph127/b/Lecture3.pdf" Homework Equations 1) Why V_{g} decreases with temperature but V_{l} increases with temperature 2) Why must phase transition happen at...
  7. J

    Solving First Order Partial Differential Equations

    Hi there, how can i solve this first order partial differential equation: grad p= (0,0,-ρg) where p=p(x,y,z,t) is pressure where ρ=ρ(x,y,z,t) is density where grad p is in three dimensional Cartesian coordinates can i just separately solve the three differential equations? ie dpx/dx...
  8. A

    Rewrite equation as two first order ODE's Help Needed

    1. rewrite the following equation as two first order ODE's x'' + 2(x^(2) - 2).x' + x = 0 Homework Equations This is what I have so far: x'' + 2(x^(2) - 2) . x' + x = 0 x'' = - 2(x^(2) - 2) . x' - x x' = y y' = -2(x(^2)-2). y - x Are these correct?
  9. B

    Solving 1st Order PDE with Initial Condition - Help Needed

    I'm trying to solve this equation: Ux + Uy + U = e^-(x+y) with the initial condition that U(x,0)=0 I played around and and quickly found that U = -e^-(x+y) solves the equation, but does not hold for the initial condition. For the initial condition to hold, I think there needs to be some...
  10. A

    Writing two first order equations in matrix form

    I have been asked to write the following two first order equations in matrix form. x' = y y' = -x I also must state that the follow on to the question asks for the only fixed point. The two first order equations came from a modified Van der pol equation.
  11. S

    Why Can't the Graph of a First Order Autonomous ODE Cross a Critical Point?

    Could someone explain why the graph of a solution can never cross a critical point?
  12. J

    Basic Theory of Systems of First Order Linear Equations

    Homework Statement [I'm going to write column vectors as the row vectors transposed, since I don't have a fancy-schmancy equation-writing program] Consider the vectors x(1)(t)=(t 1)T and x(2)(t)=(t2 2t)T (a) Compute the Wronskian of x(1) and x(2). (b) In what intervals are x(1) and...
  13. M

    Perturbation: First order correction to particle-in-box eigenstates

    First order correction to particle-in-box eigenstates for Dirac perturbation Homework Statement Calculate the first three nonzero terms in the expansion of the correction to the ground state \psi^{1}_{1} for a Dirac delta perturbation of strength alpha at a/2 (box from 0 to a). Homework...
  14. B

    Solve Nonlinear First Order Differential Equation | y'(t)=y(t)^3+f(t)

    Maybe I'm just dumb... y'(t)=y(t)^3+f(t) find y(t) Thanks...
  15. R

    Linear First Order Difference Equations (Iterative/General Method)

    Homework Statement I am almost done with a chapter all about this topic and this type of question is the only one I can't get. This is linear first order difference equations. The question is: Given the unemployment Ut equation: Ut = \alpha + \beta Ut-1 \alpha, \beta > 0 b. Suppose...
  16. L

    First Order Homogeneous Equations

    Homework Statement \frac{dy}{dx} = \frac{3xy}{3x^2+7y^2}, y(1)=1 Express it in the form F(x,y)=0 The Attempt at a Solution I'm not sure where I'm going wrong. I let v=y/x, v+x\frac{dv}{dx} = \frac{3x^2v}{3x^2+7x^2v^2}=\frac{x^2(3v)}{x^2(3+7v^2)}= \frac{3v}{3+7v^2} \Rightarrow...
  17. V

    Step response of a first order system

    Homework Statement Find the unit step response of the transfer function... a) G(s)\,=\,\frac{4}{s\,+\,4} b) G(s)\,=\,\frac{2}{0.2s\,+\,1} Homework Equations General first order step response equation... C(s)\,=\,R(s)\,G(s)\,=\,\frac{a}{s(s\,+\,a)}, where R(s)\,=\,\frac{1}{s} then do an...
  18. R

    Modeling with First Order Differential Equation

    Homework Statement A tank contains 70 kg of salt and 2000 L of water. Pure water enters a tank at the rate 12 L/min. The solution is mixed and drains from the tank at the rate 6 L/min. Find the amount of salt in the tank after 2 hours. I'm failing to figure out how to manipulate the...
  19. James889

    First order DE, using integrating factor

    Hi, I tried to solve this by using the integrating factor technique \begin{cases} dy/dt +10y = 1 \\ y(1/10) = 2/10 \end{cases} So p(x) = 10t \rightarrow e^{10t} e^{10t} \cdot \frac{dy}{dt} + e^{10t} \cdot 10y = e^{10t} This part is confusing to me, i have two different variables...
  20. G

    First Order Differential Problem

    Homework Statement Show that if a and \lambda are positive constants, and b is any real number, then every solution of the equation dx/dt + ax = b*exp(-\lambda*t) has the property that x(t) --> 0 as t --> \infty The Attempt at a Solution i tried considering the cases where a = \lambda and a...
  21. N

    Solve First order linear differential equation, initial conditions

    Homework Statement The problem is given as follows: Solve dy/dt + y = 0.5, y(t=0)=1Homework Equations The Attempt at a Solution I separate the y terms from the t terms, which gives me dy(-y+0.5)=dt I integrate both sides to get -ln(-y+0.5)=t+C C is the constant, I combine the constants from...
  22. R

    First order differential equation

    Homework Statement Solve the differential equation: Homework Equations 1+(x-x^2*e^(2y))(dy/dx) = 0 The Attempt at a Solution No idea how to approach this.
  23. Z

    Degenerate pertubation theory when the first order fails

    The basic algorithm of degenerate perturbation theory is quite simple: 1.Write the perturbed Hamiltonian as a matrix in the degenerate subspace. 2.Diagonalize it. 3.The eigenstates are the 'correct' states to which the system will go as the perturbation ->0. But what to do if the first...
  24. E

    How to Solve a First-Order Nonlinear PDE using the Method of Characteristics?

    i have to solve this equation : du/dx * du/dy = x*y u(x,y) = x for y =0 with putting this equation in the form : F(x,y,u,du/dx,du/dy) = 0 . it can be solved. But mine book does not explain how to do this, there are no examples. Can someone help me ? or any links of examples on the...
  25. D

    First Order Homogeneous Differential Equations

    Homework Statement Find the general solution of the following homogeneous differential equations: (i) \frac{du}{dx} = \frac{4u-2x}{u+x} (ii) \frac{du}{dx} = \frac{xu+u^{2}}{x^{2}} (You may express your solution as a function of u and x together) Homework Equations There are no...
  26. J

    First order perturbation theory

    The potential of an electron in the field of a nucleus is: -Ze^2/(4 Pi Epsilon0 r) r > r0 -Ze^2/(4 Pi Epsilon0 r0) r <= r0 where r0 is the fixed radius of the nucleus. What is the pertubation on the normal hydrogenic Hamiltonian? Calculate the change in energy of the 1s state to the first...
  27. K

    First order ODE, The Homogeneous Method.

    Homework Statement \frac{1}{xy} \frac{dy}{dx} = \frac{1}{(x^2 + 3y^2)} Homework Equations used the substitutions: v = \frac{x}{y} ,and \frac{dy}{dx} = v + x \frac{dv}{dx} The Attempt at a Solution took out a factor of xy on the denominator of the term on the right hand...
  28. E

    First Order Seires Solution ODE

    Homework Statement y' = \sqrt{(1-y^2) } Initial condition y(0) = 0 a) Show y = sinx is a solution of the initial value problem. b) Look for a solution of the initial value problem in the form of a power series about x = 0. Find coefficients up to the term in x^3 in this series...
  29. Y

    First order non linear partial differential equations

    Consider the following function of space and time for a propagating plane wave were nonlinear effects are included via a constant "B" u(x,t) = u[t - x/[c + Bu(x,t)]] show that u(x,t) satisfies a first order non linear PDE.
  30. A

    First order differential equation

    Hi I am looking for a firstorder differential eqaution which all of its solution, which there is a point that all of the solution goes thrugh Thank you Ariel
  31. H

    Converting an nth order equation to a system of first order equations

    Homework Statement convert y'' +x^2y'+12y=0 to a system of first order equations with initial conditions y(0)=0 y'(0)=7. Homework Equations The Attempt at a Solution first i isolate highest derivative y'' = -x^2y'-12y then i let u_1=y u_2=y' then (u_1)' = u_2 and...
  32. E

    First order differential equation

    Homework Statement y is a function of t Homework Equations y'+ky(e^-t)=l(e^-3t) The Attempt at a Solution Considering that the equation is of the form dy/dt + p(t)y =q(t) , I have been looking for an integrating factor of the form: e^{integral[p(t)dt]}, where p(t) = ke^(-t) If I...
  33. R

    Finding the time constant of a first order system

    Homework Statement I am trying to find the time constant of a thermometer that is taken from boiling water (100 deg C) and placed in ice water (0 deg C) Homework Equations See attached The Attempt at a Solution Using the equation in picture #1: I understand that I have to plot...
  34. S

    Series solution of first order ODE

    Homework Statement Find two non-zero terms of the power series solution of y' = 1 + y^2 ,y(0) = 0 by using series substitution y(x) = sum (k=0 to inf) [a][/k] *x^k Homework Equations The Attempt at a Solution First take the derivative of the power series to get y' =...
  35. A

    First Order Differential Equation

    Homework Statement Solve the following first order differential equation for x, \frac{dy}{dx} = 3xy + xy2 Homework Equations Methods: Separation of Variables, Define an Integrating Factor The Attempt at a Solution I have been staring at this question for a while now, hoping that...
  36. E

    First Order Differential Equation

    Ok so we are given a word problem discussing compound interest. In the first part of the question, we are given the equation: S(t) = (k/r)(e^rt -1) The next thing we are asked to do is calculate the value of r are given values of k, t, and S(t). The given values are k = 2000, t = 40, S(t) =...
  37. N

    How Do I Apply Binomial Expansion for x^{-1/2}(2-x)^{-1/2} Approximations?

    Homework Statement x^{-1/2}(2-x)^{-1/2} 1) approximate to lowest order in x 2) approximate to next order in x Do I apply the binomial expanion? Homework Equations The Attempt at a Solution
  38. C

    First Order ODE problem (ipod battery)

    Homework Statement Cappy McGreenhorn is looking to buy a new MP3 player. His primary concern is the battery life of the MP3 player. He has decided on either the aPod or the bPod. The percentage of battery life A(t) (after t hours) for the aPod is governed by: (100 - t2)dA/dt + 4tA = 0...
  39. P

    First order Linear PDE, Method of Characteristics

    Homework Statement x*u_{x} + y*u_{y}= 1 + y^2 u(x,1) = 1+ x; -infinity < x < +infinity Solve this parametrically and in terms of x and yHomework Equations We are supposed to solve this using the method of characteristics The Attempt at a Solution My problem is that solving the equation...
  40. K

    So, why did the author do this?

    I am confused by the following example about solving quasilinear first order PDEs. For the part I circled, the solution is just x^2 + y^2 = k where k is an arbitrary constant. To parametrize it in terms of t, can't we just put x = a cos(t), y = a sin(t) ? Here we only have one arbitrary...
  41. S

    Solving First Order Differential Equation using substitution

    Hi, Here is the equation: x+x'=5.1sin(600*t)*u(t) Our teacher gave us a hint that we should try using a substitution which is a system of sines, cosines, and looks something similar to 5.1sin(600*t)*u(t). I tried substituting: x(t)= A sin (w1*t)+B cos (w2*t)+ c cos(w3*t)*u(t)...
  42. J

    First Order Linear Differential Equation - I can't solve it

    Homework Statement Solve the differential equation dx/dt = 0.63 - (9x / 2060). Homework Equations The Attempt at a Solution I started by finding the integrating factor. So I integrated 9/2060, to get 9t/2060. Therefore e^(9t/2060) is my integrating factor. Multiply 0.63 by that integrating...
  43. R

    Proving an Autonomous First Order ODE is Bounded

    Homework Statement For the following auto. first order ode: x' = x^2 - y -1 , y' = x + x*y, show that each integral curve begins inside the unit circle remains there for all future time. Homework Equations Okay, i think what needs to be shown... define a new equation r^2 = x^2 + y^2...
  44. D

    First order DE with exponential

    I want to solve a first order differential equation on the form y'+y=e^{x} I want to separate the variables, but not sure how...Thanks!
  45. K

    First order linear ODE-integrating factor has absolute value in it

    First order linear ODE-integrating factor with absolute value?! Homework Statement Solve the ODE y' + (3/t) y = t3. 2. Homework Equations /concepts 1st order linear ODE The Attempt at a Solution Integrating factor =exp ∫(3/t)dt =exp (3ln|t| + k) =exp (ln|t|3) (take constant of...
  46. D

    Separable First Order Differential Equations: Solving y'=x√y

    I have tried to solve the differential equation y'=x\sqrt{y} like this: y^{-\frac{1}{2}}y'=x \int{y^{-\frac{1}{2}}}dy=\int{xdx} y^{\frac{1}{2}}=\frac{x^2 +C}{4} y=\left(\frac{x^2+C}{4}\right)^2 Is this the right way to solve it? Because the answer in my textbook says that the...
  47. K

    First order linear non-homogeneous PDE

    Homework Statement Find the general solution to the PDE and solve the initial value problem: y2 (ux) + x2 (uy) = 2 y2, initial condition u(x, y) = -2y on y3 = x3 - 2 2. Homework Equations /concepts First order linear non-homogeneous PDEs The Attempt at a Solution I know that the...
  48. K

    First order linear PDE-the idea of characteristic curves

    "Consider a first order linear PDE. (e.g. y ux + x uy = 0) If u(x,y) is constant along the curves y2 - x2 = c, then this implies that the general solution to the PDE is u(x,y) = f(y2 - x2) where f is an arbitrary differentiable funciton of one variable. We call the curves along which u(x,y) is...
  49. M

    First Order Homogeneous Equation

    Homework Statement (4y4-9x2y2-144)dx - (5xy3)dy = 0 Homework Equations substitute y = xv dy = dx v + dv x The Attempt at a Solution after substituting i got (4x4v4-9v2x4-14x4)dx - (5v3x4)dx.v + dv.x = (4v4-9v2-14)dx - 5v3(dx.v + dv.x) = 0 = dx(4v4-9v2-14-5v4)+dv(-5v3x)= 0...
  50. K

    What is the characteristic equation for a first order linear PDE?

    Suppose we have a first order linear PDE of the form: a(x,y) ux + b(x,y) uy = 0 Then dy/dx = b(x,y) / a(x,y) [assumption: a(x,y) is not zero] The characteristic equation for the PDE is b(x,y) dx - a(x,y) dy=0 d[F(x,y)]=0 "F(x,y)=constant" are characteristic curves Therefore, the...
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