In mathematics, a nonlinear system of equations 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 ...

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13 views

Singular solutions of a system of nonlinear 2nd order ODEs

I'm faced with the following nonlinear 2nd order system of ODEs: $$ \phi''(r)+\frac{4r^3-1}{r^4-r}\phi'(r)+\frac{r^2 h(r)^2+2r(r^3-1)}{(r^3-1)^2}\phi(r)=0, \\ ...
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34 views

Find elevator height given rope length?

This question is deceptively difficult. I feel like it's probably some classic example somewhere, but I'm not sure how to describe it in enough detail to get valid results in searching online. ...
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14 views

Least square estimation (LSE) method to solve equations

I am trying to find out the disadvantages of using least square estimation to solve non linear equations.Kindly can some on please comment on this.Thanks in advance.
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13 views

Disadvantages of Taylor series method

There is method called Taylor series method to solve non linear equations iteratively. I am interested to know ,what are the disadvantages of using this method to solve. General Idea any one please?
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1answer
11 views

Dynamically non-linear increment

I have a range of percentage like 10% to 80%. Now, I want to divide this range non-linearly in 6 parts so the 2nd part will be greater than 1st. So minimum value is 10% and it will scale up to 80% ...
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1answer
38 views

Drawing the phase portrait of a nonlinear system

Consider the nonlinear system: $$\begin{cases}\dot{x}_1=(x_1-x_2)(1-x_1^2-x_2^2),\\\dot{x}_2=(x_1+x_2)(1-x_1^2-x_2^2).\end{cases}$$ Draw its phase portrait. Solving $\dot{x}_1=\dot{x}_2=0$, we ...
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2answers
25 views

$2$-dim dynamical system IVP

Consider the IVP for the $2$-dimensional dynamical system ($X=[0, \infty )^2$) $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ for all $t \in I$, ...
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27 views

What does $D(f(\textbf{x}))$ mean

If we have a nonlinear dynamical system with $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ what do we need to do to find $D(f(\textbf{x}))$? Is it ...
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11 views

Solving a large set of equations with Newton Raphson and Gauss Jordan Elimination in VBA Excel (6x6)

What I tried to do is to solve a system of 6 equations with VBA excel. This system counts some nonlinear equations which requires Newton Raphsons method to solve and Gauss Jordan Elimination to invert ...
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1answer
34 views

Simultaneous equation with summation and square - how to solve?

$\mathbf{p}$ is a vector with dimension: $x \times 1$ $\mathbf{d}$ is a vector with dimension: $1 \times y$ $\mathbf{V}$ is a matrix with dimension: $x \times y$ $y \geq x$ $\mathbf{d}$ and ...
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2answers
41 views

Solving for the positions of vertices of 3 line segments

I have 3 line segments of lengths p,q,r joined at their ends. Let's call the vertices A, B, C, and D. Suppose D is fixed at the origin. Suppose that A is constrained to move only in the Y direction. ...
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1answer
21 views

Solving for Initial Velocity in a Projectile Motion Problem With “Wind”

The Problem Given an initial 2D position, a target 2D position, an angle, and a constant-acceleration wind vector, calculate an initial velocity that will make a projectile hit the target. Some ...
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19 views

Non-linear functional form satisfying $g(x-y) = \frac{g(x)}{g(y)}$. [duplicate]

Is there a general non-linear mapping function family, which obeys this general rule: $$g(x-y) = \frac{g(x)}{g(y)}.$$ I am looking at identifying a classification of admissible kinetics ...
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1answer
21 views

Bendixson's condition for existence of limit cycle for a nonlinear system

I encountered this example in Slotine,Lee:Nonlinear Control book. Consider the nonlinear system $$\dot{x_1} = g(x_2) + 4x_1x_2^2$$ $$\dot{x_2} = h(x_1) + 4x_1^2x_2$$ Is there a limit cycle on phase ...
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1answer
28 views

Identify if optimization problem is convex or non-convex?

I have formulated optimization problem for building, where cost concerns with energy consumption and constraints are related to hardware limits and model of building. To solve this formulation, I need ...
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1answer
21 views

Why Non Linear equations put equal to zero in Newton Raphson Mehotd

While solving non linear equations we put them equal to zero in Newton-Raphson Method.Why we do that? Any Idea?
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1answer
39 views

Numerical integration of a system of stiff ODEs starting at a singular point

Good afternoon, I have a system of $3$ highly non linear differential equations, which I have to integrate form a starting singular point $x^1=[1,1,1]$, and theoretically I have to arrive to an ...
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1answer
26 views

A 2nd order nonlinear ODE with one boundary and two algebrac equation constraints

How to solve the following nonlinear ODE with two algebraic equations and one boundary condition? ...
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9 views

An algebraic equation system and the Jacobi determinant as test for its solvability

I am trying to verify a result in a text that I am currently reading. The context is in algebra and combinatorics. However the result is obtained by using a bit of vector calculus much to my suprise. ...
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12 views

How to determine the right initial and boundary conditions of the nonlinear PDE system

The nonlinear PDE system is from a research paper in 2000. The authors solved the system by using an ordinary differential equation integrator in FortranVariable-coefficient Ordinary Differential ...
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32 views

Linearize an equation

The question is linearize the equation $\frac{du}{dt}=ru\left(1-\frac{u}{C}\right)$ about the solution $u=v^*=C$. So I let $f(u)= ru\left(1-\frac{u}{C}\right)$. I tried linearizing this. $$f'(u)= ...
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1answer
24 views

Non-linear systems convergence

Is there a way of being sure that simple iteration schemes, such as Gauss-Jacobi and Gauss-Seidel will converge for non-linear systems? I understand that for linear systems, the matrix A has to be ...
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0answers
100 views

nonlinear system: properties of solutions

I have a nonlinear system of equations which I'm struggling to solve and actually do not have much hope to find an explicit solution. I do not need to solve it; but I want to prove a property of the ...
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14 views

Reference for Inverse Scatering Transform

I am looking for a good introductory text to learn inverse scatering transformation and related topics (Lax pairs, nonlinear FT). Any pointer is much appreciated.
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1answer
13 views

Newton method norm of error is proportional to norm of residual?

Let $F(x):\mathbb{R}^n\rightarrow \mathbb{R}^n$. Newton's method is: $x_{k+1} := x_k + d_k$, where $d_k$ is computed to satisfy $F'(x_k)d_k = -F(x_k)$. If the error at the current step is $e_k = x^* ...
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1answer
35 views

Finding all roots of multivariate polynomial using Newton's method

I read that it is possible to find a solution to a nonlinear system of equations using Newton method and Jacobian matrix. But if I understood correctly, this finds just one solution, and which one ...
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1answer
45 views

Dynamical Systems: Finding Directions Around Equilibria?

I'm struggling to find the best way to determine directions about equilibria in order to draw a phase plot of this non linear dynamical system: $x' = x − xy,$ $y' = \dfrac{4}{5} − x^2 + x − y.$ By ...
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15 views

How to solve for the constants of a non-linear equation?

I don't know the correct method to solve for the constants in equations like these (when I am trying to find the solution to a trial non-homogeneous recurrence): $$a\cdot n^2 + b\cdot n^3 + c\cdot ...
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15 views

Good books on numerically solving nonlinear PDEs

I had one course in PDEs and we weren't taught numerical methods in this course, and from the books I've read on the topic it seems very hard to impossible to solve with methods like F.D or F.E to the ...
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19 views

nonlinear optimization : restricting search space to “limited” preselected value

My function is nonlinear with respect to a scalar \alpha . However, the calculation of objective function is very time consuming, making optimization also very time consuming. Also, I have to do it ...
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46 views

Solving a non-linear parametric equation

I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example: $ y^{2}(t) + y(t) = \sin(t)$ I am coming from a signal ...
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1answer
33 views

numerical solution of ode singularity

I have to solve a nonlinear ode problem: \begin{align} \dot\gamma&=-2(D+0.5A+0.5A\cos\gamma)\sin k \\ -\dot k&=(E-B-B\cos\gamma)+(2D+A+A\cos\gamma)\frac{\cos\gamma}{\sin\gamma}\cos k ...
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39 views

Differential equation system with multiplication

I have the following differential equation system, the problem being that the two unknowns $x_1$ and $x_2$ are multiplied by each other: $\left\{ \begin{eqnarray} \dot{x}_1 &=& ...
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1answer
67 views

Specific system of differential equations

I have the following system of equations: \begin{eqnarray}\frac{dx}{dt} = x(1 - x^2 - y^2) \\ \frac{dy}{dt} = y(4 - x^2 - y^2) \end{eqnarray} I want to prove that if a solutions starts (at time $t = ...
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1answer
66 views

Intersection of kernels of two matrices with nonlinear coefficients

Consider $A$ a $n\times n$ skew-symmetric matrix and $B$ a $n\times n$ symmetric matrix, the coefficients of which are (independent) nonlinear equations in $t_1,\dots,t_n$. For example, the element ...
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26 views

Is there any approach towards finding all infinite solutions of a set of nonlinear equations when the number of unknowns is more than equations?

I have 3 nonlinear equations with 4 unknowns, with some bound constraints. How can I see if there is a solution to the problem? I wonder if there is a similar approach in nonlinear equations like SVD  ...
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1answer
35 views

Can I invert this simple nonlinear equation?

Suppose $y = x * B(x)$, where $x$ is a 2d array of positive nonzero real numbers, $*$ denotes pointwise multiplication, and $B(x)$ is a blurred version of $x$, that is, $B(x)=x \otimes p$, where $p$ ...
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1answer
35 views

Solve this system of equations explicitly for $f_d$ and $\alpha$

In this system of equations $\alpha$ and $f_d$ are unknown. $\alpha$ and $C_{13}$ are complex numbers; the other parameters are all real numbers. I want to solve this system explicitly for $f_d$ ...
2
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0answers
27 views

Why does this algorithm converge?

Consider the following problem. Let $p_1, \dots, p_n \in (0,1)$ such that $\sum p_i = 1$. Let $m > 0$ such that $$ q_i := p_i + m \frac{p_i \log(p_i)}{\sum p_k \log(p_k)} < 1 $$ Suppose ...
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4answers
82 views

Solving Symmetrical Equations Algebraically

I'm doing some Cambridge STEP papers and have come across a tricky set of equations. \begin{align*} 99 &= c^3 + 6 cd^2 \tag{1} \\ 70 &= 3c^2d + 2d^3 \tag{2} \end{align*} From looking ...
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2answers
46 views

How to check the following system on existence of periodic solutions?

Let's have following system of DE: $$ \begin{cases} \dot{x} = y(1+x - y^{2}) \\ \dot{y} = x(1+y-x^{2})\end{cases}, \quad x, y \geqslant 0 $$ How to check whether this system contains periodic ...
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0answers
40 views

Solve ${\partial u\over\partial t}+u^m\left({\partial u\over\partial z}\right)^2+u^n{\partial^2 u\over\partial z^2}+u^p{\partial u\over\partial z}=0$

$$ \frac{\partial u}{\partial t} + u^m \left( \frac{\partial u}{\partial z} \right)^2 + u^n \frac{\partial^2 u}{\partial z^2} + u^p \frac{\partial u}{\partial z} = 0$$ I'm trying to solve this ...
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1answer
46 views

Solving non-linear pde with newton method

I know that to solve a nonlinear pde, you either have to linearize or you have to solve it using Newton's method. I didn't find any clue or example about how to do it with Newton's method. Can any ...
3
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1answer
32 views

Is there another way to state summations like this

I have a product looking like this: $$\omega = (aP+bQ+ \cdots)(cP+dQ+ \cdots)(eP+fQ+ \cdots)(gP+hQ+ \cdots)$$ So terms like $P,Q,R$, and so on repeat in every parentheses here, with different ...
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0answers
14 views

resolution of a nonlinear equation by using feedback

I have a nonlinear equation with one parameter and one variable.I would like to solve it by obtaining the parameter which cancels the equation. So I would like to use a PI controler for that. Is it ...
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0answers
10 views

Nonlinear Equation Systems - Existence

What are (if there are any) the rules relating a nonlinear system of equations to the existence and number of solutions ? (i.e. the role rank-determinant of coefficient matrix play in linear systems ...
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0answers
9 views

Properties of Brouwer degree

Let $X$ be a bounded open set , $f: X \rightarrow \mathbb{R}^n$ be continuous function from $X$ to $\mathbb{R}^n$ and $K$ be a closed subset of $X$ such that $y$ in $\mathbb{R}^n$ then $$\deg (f, X, ...
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1answer
38 views

How can I find the two critical points of this system of equations?

I'm currently trying to use Lagrange Multipliers to find the 2 critical points of the function $$ f(x,y,z) = \frac{1}{2}x^{2}+yz+\frac{1}{3} y^{3} - z^{2} $$ subject to $$ h(x,y,z) = x+y+z-2 = 0 $$ ...
2
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0answers
51 views

How to find the period of periodic solutions of the van der Pol equation?

The equation $y''+1.115(y^2-1)y'+y=0$ has solutions that tend towards periodic solutions and I am asked to enter the period of the periodic solutions. How can I find the period without any boundary ...
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1answer
25 views

Hot to show that system of nonlinear differential equations doesn't have periodic solutions?

Suppose we have nonlinear system of differential equations $$ \frac{d\mathbf x}{dt} = \hat{A}(\mathbf x, \mu)\mathbf x $$ How to show that it has periodic solutions?