What is Nonlinear differential: Definition and 69 Discussions

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.

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  1. W

    I Nonlinear differential equation

    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...
  2. 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...
  3. Isaac0427

    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...
  4. S

    MHB Second-Order Nonlinear Differential Equation

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  5. B

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    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...
  6. bob14

    Solving a Second-Order Nonlinear Differential Equation

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  7. Dor

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  8. S

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  9. E

    Conserved quantities in the Korteweg-de Vries equation

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  10. M

    A Nonlinear differential equation

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  11. E

    A Nonhomogeneous second order nonlinear differential equations

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  12. S

    A Nonlinear first order Differential equation

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  13. P

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  14. 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
  15. 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!
  16. A

    Trapping region for a nonlinear ODE system

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  17. 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...
  18. Last-cloud

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  19. W

    Need help solving second-order nonlinear differential eq

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  20. 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...
  21. teroenza

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    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}}...
  22. ellipsis

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  23. F

    MHB First Order Nonlinear Differential Equation

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  24. F

    MHB Need Help Solving a 2nd Order Nonlinear Differential Equation

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  25. J

    2nd Order Nonlinear Differential Equation

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  26. B

    How to solve this nonlinear differential equation

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  27. marellasunny

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  28. K

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  29. Y

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  30. P

    Linear or Nonlinear Differential Equation?

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  31. F

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  32. C

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  33. T

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    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...
  34. T

    Nonlinear Differential Equation solving help please

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  35. E

    Second order nonlinear differential equation

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  36. M

    Nonlinear differential equation

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  37. A

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

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  38. F

    Need help for solving a 2nd order nonlinear differential equation

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  39. 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...
  40. P

    First order nonlinear differential equation Help needed.

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  41. C

    Second order, nonlinear differential equation help

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  42. 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?
  43. L

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

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  44. M

    1st order nonlinear differential equation

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  45. C

    Nonlinear differential equation

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  46. F

    2nd order nonlinear differential equation

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  47. W

    How to solve this nonlinear differential equation numerically?

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  48. 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[...
  49. S

    Solving a system of nonlinear differential equations

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  50. A

    Systems of Nonlinear Differential Equations

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