Questions tagged [systems-of-equations]

This tag indicates that several equations (of some type) must all hold. Do not use alone! Use in conjunction with (linear-algebra), (polynomials), (pde), (differential-equations), (inequalities) or another tag that describes the nature of the equations being considered.

1,906 questions with no upvoted or accepted answers
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19
votes
0answers
417 views

Four squares such that the difference of any two is a square?

I. This post asks to find $4$ integers $a,b,c,d$ such that the difference between any two is a square. As mentioned by my answer, it is equivalent to finding $3$ squares such that the difference of ...
11
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0answers
402 views

How do you solve linear least-squares modulo $2 \pi$?

I have an overdetermined system of $m$ equations ($i = 1, 2, \dots, m$) $$ \sum_{j=1}^n A_{ij} \, x_j = y_i \pmod{2\pi} $$ where the $x$ coefficients are unknown, and $m > n$. This is, essentially, ...
10
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1answer
390 views

Finding a minimal set of equations that determine a variable.

I have a system of $m$ linear equations on $n$ variables, which I'm representing as $Ax=b$, with $A$ an $m\times n$ matrix representing the equations and $b$ an $\mathbb R^m$ vector representing the ...
8
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188 views

How can I guarantee the existence of a solution to this quadratic system of equations?

I have $n$ real quadratic equations and $n$ real variables, $x_i$, of the following form: $$\sum_{i\neq j} a_{ijk}x_ix_j+\sum_ib_{ik}x_i+c_k=0 \ \forall k$$ for $i,j,k\in\{1,\dots n\}$; all ...
8
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296 views

Is a finite number of quadratic equations in two variables sufficient to solve for the two variables?

Let's say I’m trying to solve a Diophantine problem in two positive integers, $y$ and $q$. Furthermore, let’s say I can derive an extremely large (read: arbitrary) number of equations of the form $$ay^...
7
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0answers
201 views

Qualitative dependence of solution to second-order matrix differential equation on eigenvalues

Suppose we have a matrix differential equation in $\vec{x}(t)=\left(\begin{smallmatrix}x_{1}(t) \\ \vdots \\ x_{n}(t)\end{smallmatrix}\right)$, such that: $$\frac{\mathrm{d}^{2}\vec{x}}{\mathrm{d}t^{...
6
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0answers
155 views

Finding a Solution to a Log-Linear System of Equations, or Showing Existence of Such a Solution

I'm trying to find the solution ($x^*_1, y^*_1, x^*_2, y^*_2$) to the following system of equations: $$ gx_1=\lambda\left(\log \frac{x_2}{1-x_2}-\log \frac{x_2 + y_2}{2-x_2-y_2}\right)\\ by_1=\lambda\...
6
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0answers
116 views

Uniqueness of a constrained system of linear equations

I would like to determine whether there exists a solution (and if so, check uniqueness) to the following system of linear equations (with respect to $\eta = (\eta_1,...,\eta_J))$: $$\begin{aligned} \...
6
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1answer
53 views

Unifying abstraction of duality between $A - B$ and $A + B$

I'm wondering whether there's an abstraction that unifies the special cases of dual or complementary equations of the form $A - B$ and $A + B$ that I've seen in math. Here are some examples: 1: Even ...
6
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0answers
261 views

How to prove existence of a solution of this determinant equation?

Let $D\in\mathbb{R}^{n\times n}$ be a real diagonal matrix where $\sum_i D_{ii}<0$. Let also $R\in\mathbb{R}^{n\times n}$ and $L\in\mathbb{R}^{n\times n}$ be real (possibly) non-symmetric (...
6
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0answers
967 views

Matrix permutation-similarity invariants

https://en.wikipedia.org/wiki/Matrix_similarity https://en.wikipedia.org/wiki/Permutation_matrix The determinant and trace (and characteristic polynomial coefficients) are well-known similarity ...
6
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0answers
229 views

Solving "ugly" equations

$k$, $c_1$ and $c_2$ are unkowns, while others are given. How can I solve the equations? $$ \begin{cases} k\left( {c_1 e^{\pi k} + \frac{c_2}{e^{\pi k}}} \right)^{(1 + 2\theta)/\theta} = - \theta^2\...
6
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0answers
124 views

Number of solutions of the linear equation

There are two positive integers $a$ and $b$ such that $a \mod b$ is not zero. We find a value $n=\lfloor a/b \rfloor+1$ where $\lfloor .\rfloor$ is the block function. We are supposed to find the ...
6
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0answers
918 views

Solving $Ax = b$ for non-negative $x$ given boolean matrix $A$ and non-negative $b$

I am trying to solve $Ax = b$ with the following properties: $A$ is a boolean (aka. logical, binary) matrix, i.e., each entry in $A$ is either $0$ or $1$ $A$ is of size $m \times n$ where $m \ll n$ ...
6
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0answers
1k views

Why does the taylor expansion of a nonlinear system of differential equations exist if it has continuous second order partial derivatives?

My textbook states that for a nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y)$$ The system is locally linear in the neighborhood of a critical point $(x_0,y_0)$ whenever the ...
6
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0answers
246 views

How to solve a particular initial-boundary value problem

I have the following initial-boundary value problem $$\begin{cases}\dfrac{\partial^2 u_1}{\partial x^2}=A_{11}\dfrac{\partial u_1}{\partial t}+A_{12}\dfrac{\partial u_2}{\partial t}\\\dfrac{\...
6
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1answer
81 views

About finding the inverse of a matrix

I am solving a linear algebra problem and this matrix came up from a system of linear equations. $A = \begin{pmatrix} 1 & 2 & \cdots & n \\ 1 & 2^2 & \cdots & n^2 \\ \vdots &...
5
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0answers
107 views

Approximate solution of $\Gamma(x+a)=k\, \Gamma(x+1)$ for $0 \leq x \leq a$

As the title says, I am looking for good approximate solution of the equation $$\Gamma(x+a)=k\, \Gamma(x+1) \qquad \text{for} \qquad 0 \leq x \leq a$$ for a given real and positive value of $a$ and $...
5
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0answers
112 views

6 linear PDE for only 3 unknowns?

Let $x \in (0,L)$, $t \in (0,T)$, and let $u_0 = u_0(x) \in \mathbb{R}^3$, $g=g(t) \in \mathbb{R}^3$, $P = P(x,t) \in \mathbb{R}^3$ and $Q = Q(x,t) \in \mathbb{R}^3$ be continuously differentiable ...
5
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0answers
256 views

Reducing to reduced echelon form in linear algebra seems to do so many useful things - what gives? Why?

What's the underlying thing about reducing a matrix to reduced echelon form that solves so many things for us? Some determinations that can be from reducing to reduced echelon form: Testing linear ...
5
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0answers
132 views

Probability to obtain no solutions for a linear system

Suppose $Ax=b$ is a linear system and A is a $n \times n$ matrix and vector $b \neq 0$. Suppose all numbers $a_i$ in $A$ and $b_i$ in $b$ belong to $\mathbb{Z}$ and suppose they are in a fixed range ...
5
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0answers
157 views

Solving a system of non linear equations. Not sure if it's unsolvable due to insufficient information

I'm not a mathematician or a math student, this is simply a problem I stumbled upon while working on a personal project, so excuse me if I seem ignorant First, I have the equation (1) $$p_{ij} = \...
5
votes
1answer
142 views

Analytic solution of the $3\times3$ symmetrical ODE system $x'_i=-x_i\cdot(x_i-\bar{x})$

Consider the following system for $x_1$, $x_2$, $x_3$ positive: $$\frac{dx_{i}}{dt}=-x_{i}\left(x_{i}-\bar{x}\right)\qquad\text{where}\ \bar{x}=\frac{x_1+x_2+x_3}{3}$$ Given a starting point such ...
5
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0answers
243 views

Large system of nonlinear equations

I am trying to solve a problem, which I find quite hard, like, headache-hard. I have to solve the following set of $M$ nonlinear equations: $$F(X)=\begin{bmatrix}f_1 (X)\\f_2 (X)\\...\\f_M (X)\\ \end{...
5
votes
1answer
203 views

Equivalence of system of nonlinear equations

Let $A\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix, $b\in\mathbb{R}^n$, $k>0$, and $g:\mathbb{R}^n\rightarrow\mathbb{R}$ be a positive function. Consider the system of nonlinear ...
5
votes
2answers
136 views

System of non-linear ODE's

do you have any suggestions to solve analytically the Non-linear ODE system $\dot x=18 x^2 y-3p x^2+6p xy$ $\dot y=18 x^2 y-6p xy $ where $p$ is a real constant. Thank you very much cheers
4
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0answers
92 views

Are the solutions of a system of real polynomial equations continuous in the coefficients?

Let $f_1(x,c_1),\ldots,f_n(x,c_n)$ be $n$ real polynomials in $n$ variables $x=(x_1,\ldots,x_n)$ with coefficients $c=(c_1,\ldots,c_n)$. Let $$ \Gamma(c)=\{x\in\mathbb{R}^n\mid f_1(x,c_1)=0,\ldots,f_n(...
4
votes
0answers
74 views

Decay of linear system with damping

Let us consider the following linear system with damping: $$ \begin{cases} u_t - u_x = -\frac{1}{2} a(x) (u+v)\\ v_t + v_x = -\frac{1}{2} a(x) (u+v) \end{cases} $$ where $a(x) = \mathbf{1}_{(-\infty,-...
4
votes
1answer
134 views

Eliminating $x,y,z$ from given set of equalities

Given that: $x-2y+z=a$, $x^2-2y^2+z^2=b$, $x^3-2y^3+z^3=c$ and $\frac{1}{x}-\frac{2}{y}+\frac{1}{z}=0$. Eliminate $x,y,z$ from given set of equations. [Hint: Use $x^2+z^2=(x+z)^2-2xz$ and $x^3+z^3=(x+...
4
votes
1answer
114 views

Gronwall lemma for system of linear differential inequalities

Let $u,v:[0,\infty)\to[0,\infty)$ satsfying the following system of differential inequalities: $$ u'(t)\leq a_1\,u(t) + a_2\,v(t) + a_0 \\[4pt] v'(t)\leq b_1\,u(t) + b_2\,v(t) + b_0 $$ for suitable ...
4
votes
0answers
54 views

Solving Vandermonde-style set of simultaneous equations

Imagine there's a set of ordered coefficients $\lambda_1>\lambda_2>\ldots>\lambda_n>0$ which I don't know. However, I know the set of relations $$ \sum_{i=1}^n\lambda_i^k(-1)^{i+1}=a_k $$ ...
4
votes
1answer
68 views

Existence of coefficient matrix for given System of Linear Equations

We know the standard form of expressing a system of linear equations in $n$ variables in $n$ equations. $$A_{n \times n} \cdot X_{n \times 1} = B_{n \times 1}$$ Where $A$ is the coefficient matrix, $...
4
votes
0answers
447 views

Trace($A$) = Trace($A^2$) = Trace($A^3$)

Suppose $A$ be an $n\times n$ matrix with real eigenvalues such that $\text{trace}(A)=\text{trace}(A^2)=\text{trace}(A^3)$. What can we conclude about the eigenvalues of A? My attempts: Considered $...
4
votes
0answers
121 views

What is the probability of exactly one negative solution in a Fibonacci system of equations?

The Fibonacci numbers denoted by $F_i$ for $i\ge1$ are $$1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,\cdots$$ where they satisfy the property $F_{i+2}=F_{i+1}+F_i$. I have listed the first $15$ ...
4
votes
0answers
103 views

Are there other values of $n$ that generate $p^2$?

I found a pattern that looks quite interesting. $$\begin{align} 2(4 + 2) + 13^3 &= 47^2 \\ 2(4 +5) + 7^3 &= 19^2 \\ 2(4 + 8) + 1^3 &= 5^2.\end{align}$$ It seems at first that if $p$ is ...
4
votes
1answer
98 views

Can we find $n$ Pythagorean triples with a common leg for any $n$?

John Leech has a nice paper entitled, "Two Diophantine birds with one stone". The two birds in question are the two systems, $$t^2−3\big(a^2, b^2, (a + b)^2, (a−b)^2\big) = p^2, q^2, r^2, s^2$$ $$u^2 +...
4
votes
0answers
174 views

A method to evaluate functional roots of $e^x$

I've an idea to find exact function $f(x)$ such that $f(f(x))=e^x$. But it involves solving complicated systems of non-linear equations, the skills for which I don't have. Here's how I intend to do ...
4
votes
0answers
50 views

System of differential equations. Recurrence relation.

Starting from physical problem of calculating the vector potential, I come up with following system of differential equations: $$ \begin{cases} \color{red}{R_{zzz}}(1-\lambda^2+z^2)+\color{red}{R_{zz}...
4
votes
1answer
131 views

Does $A$ commute with $e^{\int A \: dt}$

I have been studying the linear system of the form: $$D_tX = AX + \textbf{b}$$ Where $A$ is not necessarily constant Suppose we aim to find an integrating factor $M$ such that: $$M[D_tX - AX] = ...
4
votes
0answers
84 views

System of quadratic equations for a tetrahedron

I know the dimensions of the base of a tetrahedron and the angles between the non base sides at the apex. I want to know the lengths of the three non base sides. Let the base's corner points be $A, B,...
4
votes
0answers
257 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = kf_{...
4
votes
0answers
179 views

When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
4
votes
0answers
154 views

Is there a name for systems of equations with min and max functions included?

In a big project I'm working on, I'm running into systems of equations that look like the following: $$a = \min(b, c)$$ $$b = d^2 + a$$ $$c = \max(a + b, d)$$ Basically, nonlinear systems of ...
4
votes
1answer
310 views

Solving a set of linear equations with a total of 35 variables

I have the following complex system of equations and I need to find a solution (or any possible number of solutions): c + d + g + j + n = 2000 b + e + g + k + o + u + w + y - J = 1500 a + e + h + l + ...
4
votes
0answers
63 views

Solving a system of equations

I'm trying to prove the existence of a solution to the system of equations $$c_i = \gamma x_i + (1-\gamma) \frac{x_i^2}{\sum_{j=1}^\infty x_j}$$ for $i\in\{1,2,....\}$ where $\sum c_i=1$. I am also ...
4
votes
1answer
120 views

Solution to a simple system of quadratic equations

I am hoping to find a closed-form solution to the following system of $n$ quadratic equations: $$ x_j^2 = \sum_{i=1}^n B_{ij}x_i $$ for $j\in\{1,\dots,n\}$, where $B_{ij}\geq 0$. There is a trivial ...
3
votes
0answers
167 views

China 1997, System of Linear Equations

(source: China 1997) Problem 1. (CHINA 1997) Given that $x=2$ and $y=2$ is the solution of the system $$ ax+by=7\\ bx+cy=5 $$ Then the relation between $a$ and $c$ is a. $4a+c=9$ b. $2a+c=9$ c. $4a-c=...
3
votes
0answers
85 views

How to combine equations that are describing related physical phenomena?

A quick word about what is going on to put this question in perspective. I am trying to dock a robot onto a charger. Sometimes the robot center is not in line with the charger center. Let's call this ...
3
votes
0answers
59 views

Analysing global stability of a 1-dimensional system?

Consider \begin{align} \frac{dS}{dt} &= \mu N -\frac{\beta S I}{N} - \nu S\\[2ex] \frac{dI}{dt} &= \frac{\beta S I}{N} -\nu I \end{align} Where $N=S+I$ is the total population. By substituting ...
3
votes
0answers
77 views

Seeking the best method for solving simultaneous equations with very unique properties

Hello and thank in advance for your help. I am trying to determine the best method for solving a set of simultaneous equations with unique properties. These equations arise from a problem in ...

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