# 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. ### 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. ### 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. ### 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. ### 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} \\...
5. ### 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...
6. ### 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}...

14. ### 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. ### 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. ### 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.
17. ### 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. ### 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...
19. ### 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...
20. ### "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. ### 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}}...
22. ### 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...
23. ### 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]
24. ### 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!
25. ### 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...
26. ### 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.
27. ### 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...
28. ### 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...
29. ### 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...
30. ### 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...
31. ### 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...
32. ### 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...
33. ### 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...
34. ### 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...
35. ### 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...
36. ### 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...
37. ### 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...
38. ### 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
39. ### 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. ### 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...
41. ### 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)
42. ### 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. ### 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?
44. ### 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...
45. ### 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...
46. ### 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...
47. ### 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?
48. ### 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. ### 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...
50. ### Systems of Nonlinear Differential Equations

Hi, I trying to solve a system of Nonlinear Differential Equations. I'm using Runge-Kutta on the Differential equations and Newton Method for the system. I have some doubts in how to create the JAcobian to the differential equations. Could somebody help me, please? Thank you, Aline