Nonlinear differential Definition and 70 Threads

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.
In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.
As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.
Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:

Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.

View More On Wikipedia.org
Recent contents Top users
  1. LazyPhysicist1

    I Solution To Photon Trajectory Equation

    Using the null geodesic and the Schwarzschild metric, this differential equation for photon trajectory near a mass can be derived, where u is r_s /2r: Though this nonlinear ode is fairly easy to approximate (which I already have), I'm looking for an analytic solution or an approximate...
  2. W

    I Why Is a Differential Equation Called Nonlinear?

    hi, i am working on nonlinear differential equation- i know rules which decide the equation to be nonlinear - but i want an answer by which i can satisfy a lay man that why the word nonlinear is used. it is easy to explain nonlinearity in case of simple equation i.e when output is not...
  3. A

    Solving coupled equations analytically

    The equation is as attached where, - α, β and γ are constants - i1 and i2 are the variables. Also attached, is my attempt and where I stuck at. If anyone has an idea how to convert this into Bernoulli’s form, please I really need help. If there are any other ideas please let me know too...
  4. I

    I An interesting Nonlinear Differential Equation

    That's pretty much it. If there is a very basic strategy that I am forgetting from ODEs, please let me know, though I don't recall any strategies for nonlinear second order equations. I've tried looking up "motion of a free falling object" with various specifications to try to get the solution...
  5. S

    MHB Second-Order Nonlinear Differential Equation

    Hi there can someone please help me with this differential equation, I'm having trouble solving it \begin{cases} y''(t)=-\frac{y(t)}{||y(t)||^3} \ , \forall t >0 \\ y(0)= \Big(\begin{matrix} 1\\0\end{matrix} \Big) \ \text{and} \ y'(0)= \Big(\begin{matrix} 0\\1\end{matrix} \Big)\end{cases} \\...
  6. B

    First-order nonlinear differential equation

    Homework Statement: first order non linear equation Homework Equations: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4 a,b,z,v,u are constant t0=0 , T=T0 Hi, i need find an experession of T as function of t from this first order nonlinear equation: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4 a,b,z,v,u are constant...
  7. bob14

    Solving a Second-Order Nonlinear Differential Equation

    Homework Statement Hi, I'm trying to calculate the formula for the position vs. time of a rocket landing from an altitude of 100km. I'm neglecting a lot of forces for simplification but basically, I want to solve ##F_{net} = Drag - mg##. Homework Equations Drag Force: D = ## \frac {C_dAρv^2}...
  8. Dor

    A Numerical solution for a two dimensional 3rd order nonlinear diff. eq.

    I'm a bit lost in all the numerous methods for solving differential equations and I would be very grateful if someone could point me to some direction. I want to solve the following boundary conditioned differential equation: $$a_1+a_2\nabla f(x,y)+a_3\nabla f(x,y)\cdot \nabla^2...
  9. S

    What is 'phase space in chaos theory and nonlinear dynamics?

    The term 'phase space' is often used in the study of nonlinear dynamics.What is it.
  10. E

    Conserved quantities in the Korteweg-de Vries equation

    Homework Statement Consider the Kortweg-de Vires Equation in the form $$\frac{\partial \psi}{\partial t}+\frac{\partial^3 \psi}{\partial x^3}+6\psi\frac{\partial \psi}{\partial x}=0$$ Find the relation between the coefficients ##c## and ##d## , such that the following quantity is conserved...
  11. M

    A How Can Taylor Series Expansion Help Solve This Nonlinear Differential Equation?

    ρCp (∂T/∂t) + k (∂2T/∂x2) = exp(-σt2)exp(-λx2)φo i have this equation... i was thinking of taylor series expansion to solve it and make it easier... any ideas on how to solve?
  12. E

    A Nonhomogeneous second order nonlinear differential equations

    Hello every one, I have an equation related to my research. I wonder if anyone has any suggestion about solving it? y''+y' f(y)+g(y)=h(x) thanks
  13. S

    A Nonlinear first order Differential equation

    I need to solve the well known momentum equation in 3D cylindrical coordinates: ρ(∂v/∂t +(v.∇)v)=A where A and the velocity v are both local vector variables. I am actually looking for the stationary solution to the equation, i.e. no ∂/∂t term) I have tried evolving the velocity and tried...
  14. P

    Nonlinear differential equation

    Homework Statement I need to solve the follwing differential equation $$(\frac{df}{dt}) \dfrac{4 n e^{4nf(t)}-9n e^{2nf(t)} (\frac{df}{dt})^2 + e^{2nf(t)} r^2 \frac{d^2f}{dt^2}+5n (\frac{df}{dt})^4 r^4 - r^4 \frac{d^2f}{dt^2} (\frac{df}{dt})^2}{-e^{2nf(t)}+ (\frac{df}{dt})^2 r^2}=0 $$...
  15. E

    Solve coupled nonlinear differential equations

    Good evening I have these coupled equations and was wondering if there is any chance solving them analytically. If not, how would you approach it numerically? (shown in attachment) Thank you very much
  16. 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!
  17. A

    Trapping region for a nonlinear ODE system

    I need to find a trapping region for the next nonlinear ODE system $u'=-u+v*u^2$ $v'=b-v*u^2$ for $b>0$. What theory i need to use or which code in Mathematica o Matlab could help me to find the optimal trapping region.
  18. H

    System of nonlinear differential equations

    Hello I have a system of differntial equations: dx/ds = sin(p) dy/ds=cos(p) dp/ds = k dk/ds = -1/EI(s)*(k*dEI/ds+f*sin(p)) x(0)=y(0)=p(0)=p(L)-pl = 0 These are nonlinear differential equations. I should use some sort of nonlinear finite difference. But I do struggle to setting up the finite...
  19. Last-cloud

    Finite difference method nonlinear PDE

    i want to solve a nonlinear PDE with finite difference method ,but using just discretization like in linear PDE , it will lead to nowhere , what's the right way to use FDM to solve nonlinear PDE or could someone provide me with book's titles or articles that can help me solving a nonlinear pdf...
  20. W

    Need help solving second-order nonlinear differential eq

    Hello, I'm a doctoral student in civil engineering. In my research I came across a differential equation for the net force acting on an object as it impacts a granular medium at low velocities. z'' + a[ z' ]^2 + b[ z ] = c Where a, b, and c are all constants I believe that this equation will...
  21. M

    "State of the Art" in nonlinear differential equations?

    In my introductory ODE class we have focused mostly on linear differential equations. I know that nonlinear differential equations are much harder to solve, and I am wondering what exactly the "state of the art" methods are for dealing with them, or also what recent developments have been made...
  22. teroenza

    Series solution to nonlinear differential equation

    Homework Statement By truncating the differential equation below at n=12, derive the form of the solution, obtaining expressions for all the ancoefficients in terms of the parameter \lambda .Homework Equations The ODE is: \frac{\mathrm{d^2}\phi }{\mathrm{d} x^2} = \frac{\phi^{3/2}}{x^{1/2}}...
  23. ellipsis

    Solving nonlinear differential to emulate Newtonian gravity

    This is part of a personal project... I've recently become addicted to modeling various physical systems from scratch, such that I find explicit solutions of position as a function of time, and graph em. But I've hit a glass ceiling trying to find an analytic solution to the 1-dimensional...
  24. F

    MHB First Order Nonlinear Differential Equation

    Hi, I need help solving this ode, when I try to solve it I end up with a big crazy answer and I believe it should be simpler. (dy/dx)^2=((ay^4)/2)-(a+1)y^2+1 y(0)=0, y'(0)=1 and a is within [0,1]
  25. F

    MHB Need Help Solving a 2nd Order Nonlinear Differential Equation

    Solve the 2nd order nonlinear differential equation, with initial conditions y(0)=0 and y'(0)=1 y''=2ay^3-(a+1)y with a within [0,1] I am pretty much lost on how to go about solving this. It would be greatly appreciated if someone could point me in the right direction on this. Thanks!
  26. J

    2nd Order Nonlinear Differential Equation

    Hi everyone, I need some help to solve this differential equation. The question states "Use the perturbation or multiple scale method to find the third-order approximate solution for the following system: diff(x(t), t, t)+w^2*x(t)*(1+epsilon*x(t)^2) = 0 " Currently, I am still...
  27. B

    How to solve this nonlinear differential equation

    dy/dx=2x+y^2 By the way, methods of solving linear differential equation are useless, such as integrating factor and Bernoulli method.
  28. marellasunny

    Number of parameters for a nonlinear differential equation

    Take for example a system \frac{dx_i}{dt}=(x_i,t,a,b,...) i-number of state equations. What would be the maximum number of parameters permitted for this system of non-linear differential equations? Is it finally determined by the solution space?Is there a criteria for number of...
  29. K

    Motion in Nonlinear Differential Equations

    Homework Statement How do you derive the time-dependent velocity equation for motion along a curve, such as a skateboarder on a half pipe? For the sake of abstraction, I ask myself the following: A uniform sphere of mass m and radius r is set free from the top edge of a semicircle half pipe...
  30. Y

    How to go about solving this first-order nonlinear differential equation?

    I saw this post at stackexchange: I ran across this post when trying to solve a homework problem. But I have no idea how he got that solution for that. When I use the Euler-Lagrange, I get this diff eq below. Here is the simplest form I have managed to get it in...
  31. P

    Linear or Nonlinear Differential Equation?

    Homework Statement Is this differential equation linear or nonlinear? Assume that y' means dy/dx. Homework Equations 1. Homework Statement [/b] Is this differential equation linear or nonlinear? Assume that y' means dy/dx. Homework Equations \sqrt{xy'+2x2}=5 The Attempt...
  32. F

    Nonlinear differential equation (Laplace transform?)

    Hi, Part of my research, I nondimensionalized an ODE to eventually arrive at this form: sin(τλ) = q^((n+2)/(n+1)) + κq' + q'' where q' = dq/dτ The problem is of course the nonlinear q^n. n is an integer greater than 0. Is there a Laplace transform for this? Or what solutions are there for...
  33. C

    Euler-Bernoulli Beam Theory and Nonlinear Differential Equations

    I've been reading through my mechanics of materials textbook recently, notably in regard to the section on the deflection of beams. The well regarded Euler-Bernoulli beam theory relates the radius of curvature for the beam to the internal bending moment and flexural rigidity. However the theory...
  34. T

    Coupled Nonlinear Differential Equations

    Hey, I need your help to solve the following set of coupled differential equations numerically. dn(t,z)/dt=I^5(t,z)+I(t,z)*n(t,z) dI(t,z)/dz=I^5(t,z)-α(n(t,z))*I(t,z) where I(t,0)=I0*exp(-4ln2(t/Δt)^2) and n(t,0)=0 and n(-certrain time,z)=0. Some constant parameters I did not show...
  35. T

    Nonlinear Differential Equation solving help please

    Homework Statement Consider the following differential equation: x^{2}\frac{dy}{dx}=x^{2}-xy+y^{2} State whether this equation is linear or nonlinear and find all it's solutions Homework Equations I think that the Bernoulli differential equation is relevant, but I'm not sure...
  36. E

    Second order nonlinear differential equation

    Is it possible to solve the following differential equation analytically? y''(x) = A - B [exp(y(x)/C) - 1] where A, B and C are constants. Thank you...
  37. M

    Nonlinear differential equation

    Homework Statement In one problem I had got to this equations, but I was not able to solve it, because I'm actually on high school. The equation : d^2/dt^2(x) = -h*g/(h+x) I tried use separation of variables but I was not able to use the chain rule. Can anybody show me the steps...
  38. A

    MATLAB MATLAB: How to solve a system of Nonlinear Differential Equations?

    Hello to everybody, I'm very new to solving ODES and equations with MATLAB. I have been asked to solve a system of nonlinear equations for simulating the growth of a solid tumor. Assuming that we have the 5 unknowns which are dxd arrays: f,g,m,p and n. f(x,t) is the volume fraction of tumor...
  39. F

    Need help for solving a 2nd order nonlinear differential equation

    Hi, I need some help to find the analytical solution of the following DE: x" - k x/x' = at + b, with x' = dx/dt and x" = d(dx/dt)/dt Any kind oh help or advices on where I can find some useful resources are really appreciated. Thank you
  40. P

    First order nonlinear differential equation

    Homework Statement Find the orthogonal trajectories of the given families of curves. x^2 + y^2+2Cy=1 Homework Equations The book has covered homogeneous and separable methods.The Attempt at a Solution To find the orthogonal trajectories, we simply find the curves whose tangents are...
  41. P

    First order nonlinear differential equation Help needed.

    Hi All, I have been trying to solve following nonlinear differential equation, but I couldn't solve it. 0 = a*[f(t)]^{z/(z-1)} + (-t+C)*f(t) + b*[df(t)/dt] where a, b and C are constants and 0< z<1. Could you please help me how to solve this nonlinear differential equation? I would...
  42. C

    Second order, nonlinear differential equation help

    I've been trying to solve this one for a while. My professor wasn't even sure how to do it. Any suggestions? y''+t*y+y*y'=sin(t)
  43. C

    Nonlinear differential equation problem.

    Homework Statement The following equation turned up while I was trying to make an integral stationary in a 'calculus of variations' problem. y^{\prime}(x)^2 + 1 = y^{\prime\prime}(x) y(x) How would one go about solving this nonlinear equation?
  44. L

    Solving a Nonlinear Differential Equation: y+4y^2=(y^(4)+x)y

    Homework Statement y+4y^2=(y^(4)+x)y', IC: y(1)=1 Homework Equations ? The Attempt at a Solution Ive tried to figure out a substitution that will make this linear, and i can't seem to figure one... I am unsure of how to approach this?
  45. M

    1st order nonlinear differential equation

    this is the problem: x2y' = (2y2 - x2) here's what i have done so far: dy = (2y2/x2 - 1)dx (2y2/x2 - 1)dx - dy = 0 i used the substitution y = xv then found an integrating factor and got dx/x - dv/(2v2 - 2v - 1) = 0 but i am stuck at this point.. i know ln(x) + C is the first part...
  46. C

    Nonlinear differential equation

    Homework Statement y''+4\left(y'\right)^{2}+8=0 Homework Equations u=y'? The Attempt at a Solution I don't really know where to start, do I use u=y' substituted? So, y''=u*(du/dy)? That leads to u\frac{du}{dy}+4u^{2}+8=0 I don't think this is correct, since it leads to...
  47. F

    2nd order nonlinear differential equation

    Hello everybody, could you please direct me how to solve this nonlinear differential equation analytically, so by mathematica or matlab? I really need to solve it for my research project, so please help me du/dx=d/dx[a*u^(-1/2)*du/dx]-n*u^(3/2)*(u-c)/b boundry conditions are: u(0)=b+c...
  48. W

    How to solve this nonlinear differential equation numerically?

    -f''-(2/x)f'+(2/x^2)f+f^3-f=0 the boundary condition is f(x=0)=0 and f(x=\infty)=1 how to solve f numerically?
  49. Q

    A system of 1st order nonlinear differential equations

    Hello, Can you give some suggestions to solve the following system of 1st order nonlinear differential equations? Thank you. \[ \begin{array}{l} u'(t) = Au^2 (t) + B(t)u + C(t) \\ u(t) = \left[ {\begin{array}{*{20}c} {x_1 (t)} \\ {x_2 (t)} \\ \end{array}} \right] \\ A = \left[...
  50. S

    Solving a system of nonlinear differential equations

    Homework Statement I'm working on a problem for my robotics class and could really use some help. I am suppose to be modeling a planar scara manipulator and have managed to come up with two nonlinear differential equations that describe the system; they are shown below. \Theta_{1} and...
Back
Top