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.

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40 views

Solving complicated system of equations

Suppose I have the general equation like $$ a_j = -(k+\sum_{i\neq j}^n\alpha_iba_i)^{-1},$$ where $j=1,\cdots,n$ and $\alpha_i,k$ are constants for $i=1,\cdots,n$ and $\sum_{i=1}^n\alpha_i=1$ Is there ...
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2answers
52 views

How to know if a system of equations of the form $ y_i = \sum_{j=0}^{n} c_j e^{jx_i}$ is solvable

I was working on a problem and faced a this system of equations ($y_i$ and $x_i$ are givens) $$ y_i = \sum_{j=0}^{n} c_j e^{jx_i} \quad0 \le i \le n$$ is there a way to determine this system is ...
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1answer
12 views

LSQR method for solving a linear equation with positive value constraint for one column of the solution

I am solving an overdetermined sparse linear problem (Ax= B) using a C code. The code is using the LSQR method to find the solutions. There are 6 unknowns for every equation. One of the solutions is a ...
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15 views

On the solution space of linear matrix equations.

Consider a linear matrix equation $$ Y = \sum_{i} A_i X B_i^T = \sum_i (B_i\otimes A_i) \cdot X = T\cdot X $$ When does the solution space admit a basis consisting only of rank-1 matrices?
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10 views

Analysis of a particular instance of a non-linearsystem of equations

Consider a vector space $V \in \mathbb{R}^n$. Given a set of $\frac{1}{2}n(n-1)$ square matrices $\{\boldsymbol{A}^{(ij)}\}_{ij}$ (where $(\boldsymbol{A}^{(ij)})^T = \boldsymbol{A}^{(ji)}$), how do we ...
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27 views

Iterative method of solving systems of first order ODE's

Method I developped an iterative method of solving equations of the following form: $$\mathbf{x}'(t)=A(t)\mathbf{x}(t)$$ The attempted solution proceeds as follows: $$\mathbf{x}'=A\mathbf{x}\\ \mathbf{...
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41 views

Fundamental Solution of system of Ode: $\dfrac{1}{x}$

I have given the system: $$\left(\begin{array}{c} {y_1}'\left(t\right)\\ {y_2}'\left(t\right) \end{array}\right) = \left(\begin{array}{cc} -\frac{2}{t} & \frac{1}{t}\\ \frac{3}{t} & 0 \end{...
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19 views

Using $u(t,\gamma(t,x))=\gamma_t(t,x)$ to find various partial derivatives of $\gamma$ in terms of $u$

I have two functions $u:\mathbb{R}\times S^1\rightarrow S^1$ and $\gamma:\mathbb{R}\times S^1\rightarrow S^1$ related via this composition $u(t,\gamma(t,x))=\gamma_t(t,x)$ (call this equation $(1)$), ...
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17 views

Constants for inhomogeneous system of ODE

For a first order inhomogeneous ODE we've got a formula $$ \displaystyle{y(x) = Y(x)\,\left[c_0+\int_{x_0}^{x}Y^{-1}(x)\,b(x)\,\mathrm{dx}\right]}, $$ where $Y(x)$ is that fundamental matrix ...
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2answers
48 views

Solving ODE-systems with Matrix exponential is wrong?

Originally I've learned that the solution of a systems of coupled ODE: $$\underbrace{\left[\begin{array}{cc}{y_1}'(x)\\ \vdots \\{y_n}'(x)\end{array}\right]}_{y'(x)}= \underbrace{\left[\begin{array}{...
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How to find a polynomial relation of 1 variable from 4 simultaneous equations?

I have $4$ simultaneous equations derived from Saha equation to solve for plasma species evolution as temperature changes, $4$ variables $x,y,z,w$ represent number density of electron, neutral atom, ...
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Prove that every solution of $\dot{y}=B(t)y$ is bounded on $[\beta, \infty)$. For anu solution $x$ of $\dot{x}= A(t)x$.

Suppose A satisfies the conditions in $\dot{x}= A(t)x$ and $B(t)$ is a continuous real $n \times n$ matrix for $t \geq \beta$ with $\int^{\infty}_{\beta}|A(t) -b(t)| < \infty$. Prove that every ...
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31 views

Equation of Motion for a Point Vortex System. Need Help Solving 4 Simultaneous ODEs

Background and Statement of the Problem We consider the problem in $\mathbb{R^2}.$ As the title states, we are interested in the following equation involving the Hamiltonian of the point vortex system,...
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1answer
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Matrices: Using the row echelon [duplicate]

This is my system of linear equations $$ \left\{ \begin{array}{c} 1x_1+2x_2-1x_3=2 \\ -3x_1+1x_2-3x_3=1\\ 4x_1+ax_2-4x_3=b \end{array} \right. $$ My Rank matrix looks like this: $$ \begin{...
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Given the order relationship between multiple variables with sum 1 and their entropy, how to prove that the solution is unique. [closed]

Given the following four conditions: $x_i\ge 0,\ i=1,2,\cdots, n$ $\sum_{i=1}^n x_i = 1$ $x_i \le x_{i+1}, i=1,2,\cdots,n-1$ $\sum_{i=1}^{n} x_i\log x_i = t$, where $t$ is any value that $\sum_{i=1}^{...
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1answer
43 views

System of equations from given solution

I'm stuck at part b) of the following problem: a) Find a $2\times 3$ system $Ax = b$ whose complete solution is $$\vec x = \begin{bmatrix} 1\\ 2\\ 0 \end{bmatrix} + w \begin{bmatrix} 1\\ 3\\ 1 \end{...
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33 views

To show exactly one system has a solution [closed]

I would like to have some hint over this problem. Let A be $m\times n$ real matrix and $b\in R^m.$ Then show that exactly one of the following systems has a solution. $A\bar{x}=b$ $A^T\bar{y}=0$, ...
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1answer
26 views

Solving a system of differential equation for a thermodynamic problem

I would like to reproduce the results from a publication (https://doi.org/10.1007/s00161-015-0415-8) for verification. For this I want to solve the following system of differential equations: $$\dot{x}...
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1answer
33 views

Calculate first order differential equation eigenvectors

I have a system of first order defferential equation that has the coefficient matrix in the picture. I calculated the eigenvalues Eigenvalues: $+i, -i$ $$ Q = \begin{bmatrix} 0 & -1 & 0 &...
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1answer
62 views

Is this system of equations solvable?

I have derived a handful of equations to describe how it may be possible to find the angle of a rectangle inscribed inside of another rectangle in two different states. The equations I have come up ...
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1answer
44 views

What does it mean when the bottom row of a reduced row echelon form is all zeros with a 1 at the end?

I've just started my linear algebra course, and one question has me reduce an augmented matrix A of the form $A=\begin{bmatrix} n_{11} & n_{12} & n_{13} & a\\ n_{21} & n_{22} & n_{...
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36 views

Existence of Solution to Newtons dynamics. [closed]

Hi I'm wondering how to obtain the existence of solutions to the following problem of particle dynamics : Let $k,d\in \mathbb{N}$, $k\leq d-1$, does there exist solutions, atleast on small time ...
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2answers
46 views

Solving a system with circle equation [closed]

I am stumped by this. How do I solve algebraically, without graphing? I've tried to solve for x in the second equation and substitute into the first. I'm not sure how else to do this. It just ends up ...
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1answer
40 views

Given a trigonometric equation, use the identity and unit circle to solve $ 2 \tan (\theta) - \cot (\theta) -1=0$

Solve the equation : $$ 2 \tan (\theta) - \cot (\theta) -1=0$$ I solved this equation and got the value of $\tan (\theta)$ as $\frac{-1}{2}$ and $ -1 $ , I want to write its general formula .Kindly ...
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7 views

Bound for the number of integral solution

Is there a way to upper bound the total number of solution of equation $2x^2+y^2+32z^2=n$? I am actually trying to find the total number of integral solution of this equation, and I think getting and ...
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34 views

If two matrices are row equivalent, will there transpose be row equivalent?

Let's suppose I have two matrices A and B of same dimensions. Given is A and B are row equivalent. Will transpose (A) be row equivalent to transpose (B)? If yes then prove it.
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22 views

Two variables modulo equation [duplicate]

How do we solve an equation of this form: $ax + by = z$ for integers. For example: $45x - 37y = 25$ My try: $45x \equiv 25 (mod 37)$ $9x \equiv 5 (mod 37)$ $45x \equiv 5 + 37 \cdot 4 (mod 37)$ $9x \...
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1answer
23 views

Can I use $||J||_2$ as a “gradient descent” for the system $J = b - Ax$?

One quick question. Let's assume that I want to solve $Ax = b$ but I want to do that in a special way. My idea is that I first find the difference between $b$ and $Ax$, we call it $J$. $$J = b-Ax$$ ...
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0answers
31 views

Find a system of differential equation for N given vectors as its answers [closed]

Can each nonlinear N given vectors be the solutions of a system of differential equations?in another word, can we find a system of N first order differential equation that have this N given vectors ...
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2answers
41 views

Solving System of Nonlinear Equations Symbolically

I'm trying to solve the following system of three nonlinear equations for the variables $x, y$ and $P$. \begin{equation} \begin{split} 0&=r-\theta P-ax -by-cP-q_{1}E_{1}\\ 0&=dx-\eta -eP-q_{2}...
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1answer
33 views

How to solve $1-\sum_{i=1}^Nx_i=a_nx_n^{1/n}$ given $a_n$

I'm trying to solve (numerically) the following system of coupled equations: $$1-\sum_{i=1}^Nx_i=a_nx_n^{1/n}$$ given $a_n$, I need to find the $N$ $x_n$. I tried to write it as a matrix equation but ...
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16 views

Equilibrium of system of differential equations

Consider a system ${\underline{x}}'(t)=f(\underline{x}(t))$, which has an equilibrium point $\tilde{x}=\underline{x}(\tilde{t})$, i.e. $\underline{x}'(\tilde{t})=\underline{0}$. How does it follow ...
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1answer
54 views

What is the equation for $y$ in the system $\sin(2 * \pi /y)=0$ and $\sin( \pi y) = 1$? [closed]

What is the equation for $y$ in this system of equations? $\sin(2 * \pi /y)=0$ and $\sin( \pi y) = 0$
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29 views

System of three equations in three variable having degree 1,2&3 respectively.

Can we solve a system of three equations in three variable , with the equations having degree 1,2 and 3 ??? Consider $$P(x,y,z)=a $$ $$Q(x,y,z)=b $$ $$R(x,y,z)=c $$ where $P,Q,R $ are polynomials of ...
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1answer
26 views

How do you represent a system of equations with arbitrary rows that follow a pattern?

I have a system of equations where each equation can be expressed as: $\sum_{i=1}^n c_iy_i^{k}(t) = \bar{y}_k$ And $k$ represents the $k^{th}$ equation. I am tempted to succinctly write the lot as: $\...
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0answers
38 views

Showing that roots common to $z^m-1=0$ and $z^n-1=0$ where $\gcd(m,n)=d$ are given by all roots of $z^{d}-1=0$ [duplicate]

I cannot seem to get this proof right, which is to show that roots common to $z^{m}-1=0$ and $z^{n}-1=0$ where $\gcd(m,n)=d$ is given only by all the roots of the equation $z^{d}-1=0$. For starters, ...
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2answers
223 views

Checking and proving unicity of solution of a system of equations

Consider the following system of equations: $$\prod_{j=1}^K\alpha_j^{R_j} p_i+\prod_{j=1}^K(1-\alpha_j)^{R_j}(1-p_i)=y_{i,(R_1,\cdots,R_K)}$$ for each $i\in\{1,\cdots,I\}$ and each $(R_1,R_2,\cdots,...
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1answer
21 views

Doubt about Higher Order Differential Equations (Linear and Non Homogeneous)

Theory that I've been taught: Algorithm to solve the following equations: \begin{equation*} x^{(n)} = \sum_{j=1}^{n}a_jx^{(n-j)}+b(t) \end{equation*} First, we associate the linear homogeneous ...
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1answer
42 views

Solving system of equations involving complex exponentials

I am stuck while trying to solve the following system of equations involving complex exponentials: $$x_1 e^{j\theta_1 m_1} + x_2e^{j\theta_2 m_1} = z_1$$ $$x_1 e^{j\theta_1 m_2} + x_2e^{j\theta_2 m_2} ...
3
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1answer
37 views

Differential equation: $2x^2y'=y^2(2xy'-y)$

Solve the differential equation: $$2x^2y'=y^2(2xy'-y)$$ I tried to convert it to the form of a total differential equation. $$\begin{array}{lrl} &2x^2y'&=y^2(2xy'-y)\\ \Leftrightarrow&y'(...
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0answers
76 views

When can an abelian group be expanded to satisfy some equations and not others?

Given an abelian group $G$ and a system of equations A, $$a_1 = n^1_1 x_1 + \cdots + n^1_r x_r$$ $$\vdots$$ $$a_s = n^s_1 x_1 + \cdots + n^s_r x_r$$ with $a_i \in G$ and $n^i_j \in \mathbb{Z}_{\geq 0}$...
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1answer
60 views

Solving the Compton Scattering

This question refers to the Compton Scattering. We have an elastic impact between a photon and an electron, so conservation of $E$ and $\vec{p}$ in a 2D plane: $$\begin{cases}E^i_p+E^i_e=E^f_p+E^f_e \\...
1
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1answer
62 views

Find all ordered pairs of integers$(x,y)$ which satisfy the equation $2(x^2+y^2)+x+y=5xy$

Find all ordered pairs of integers$(x,y)$ which satisfy the equation $2(x^2+y^2)+x+y=5xy$ We can transform this equation as $$\begin{equation} 2(x^2+y^2-2xy)+x+y-xy=0\\ \implies2(x-y)^2+x+y-xy=0\\ \...
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0answers
51 views

An “interwined” system of modular equations

Well I was just solving a problem that has an alternative (non-elementary) solution using the elliptic curves. Seeking for an elementary solution, I had to find all possible values of $xy$ such that ...
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0answers
17 views

Finding maximum in and exact solution to discrete SEIR model numerical solution

I am trying to make a discrete SEIR model with the I category partitioned into two; Iasymptomatic and Isymptomatic, thus it is a SEIIR model. I solve it numerically using Euler method (in python). I'...
1
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2answers
105 views

Question from pathfinder for Olympiad mathematics 2 [duplicate]

If $p$, $q$, $r$ are the real roots of equation $x^3-6x^2+3x+1=0$, determine the possible value of $p^2q+q^2r+r^2p$. My Attempt: $p+q+r=6 (1)$ $pq+qr+pr=3 (2)$ $pqr=-1 (3)$ Multiplying (1) ...
8
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2answers
185 views

What are the conditions for a system of equations to have a solution in some abelian group extension?

Given an abelian group $G$ and a system of equations $$a_1 = n^1_1 y_1 + ... + n^1_r y_r$$ $$...$$ $$a_s = n^s_1 y_1 + ... + n^s_r y_r$$ for $a_i \in G$ and $n^i_j \in \mathbb{Z}_{\geq 0}$, under what ...
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0answers
40 views

Find a basis for $U_{1} +U_{2}$

Set $V=ℝ^{6}$ Let $U_{2}$ be the solution space of the system \begin{cases} 3x₁ +2x₂ -x₃ +4x₄ +x₅ -x₆ &= 0\\ x₁ +2x₃ +x₄ -x₅ -x₆ &= 0\\ 2x₁ +4x₂ -10x₃ +4x₄ +6x₅ +2x₆ &= 0 \end{cases} ...
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0answers
18 views

Possible values of coeficient 'a' in which elimination breaksdown,temporarily and permanently. [closed]

I need to find 3 values of 'a' for which the given system of equations, the elimination breakdown, temporarily or permanently. au + v = 1 4u + av = 2 Breakdown at the first step can be fixed by ...
8
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2answers
178 views

Solve system of equations $3(x+\frac{1}{x}) = 4(y + \frac{1}{y}) = 5(z+\frac{1}{z})$, $xy+yz+zx = 1$

Find all $x,y,z>0$ such that $$3(x+\frac{1}{x}) = 4(y + \frac{1}{y}) = 5(z+\frac{1}{z})$$ $$xy+yz+zx = 1$$ The only solution should be $x=\frac{1}{3}$, $y = \frac{1}{2}$, $z=1$. There is a way to ...

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