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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3answers
67 views

Having trouble simultaneous trigonometric equations

I'm racking my brain trying to solve some formulae that I will need to implement into a program I'm making. The program is based around statics of a particle, as in that all forces acting on the ...
0
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0answers
14 views

Independence of a binary form and its Hessian

Let $f\equiv f(X,Y):=\sum_{i=0}^{n}a_iX^{n-i}Y^i$ be a binary form of degree $n\geq3$ with coefficients over $\mathbb{C}$ and no repeated roots in $\mathbb{C}^2$ (up to scaling). The Hessian of $f$ is ...
1
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0answers
75 views

Is there a way to solve these equations?

I am trying to solve a problem that involves solving these three first order differential equations. $$ x'(r)= \,\frac{2 c \,x(r)-2 c+g \,r^2 \epsilon +h r^2-4 r^2 \epsilon x(r) z''(r)-r^2 \epsilon ...
0
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0answers
8 views

Use a uniformed scale to represent multiple linear/exponential equations

I need to allow a user to select a date value ranging from $0$ to $50$ years from a linear slider. There are $8$ uniformed intervals of $(0m, 3m, 9m, 1y, 5y, 10y, 30y, 50y)$. I need a formula to ...
0
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1answer
46 views

Is there any algorithm to find all the solutions of the following special linear Diophantine system?

Consider the following system. 1) $a_{11}x_1 + a_{21}x_2 + \cdots + a_{m1}x_m=d_1$ 2) $a_{12}x_1 + a_{22}x_2 + \cdots + a_{m2}x_m=d_2$ $\vdots$ n) $a_{1n}x_1 + a_{2n}x_2 + \cdots + a_{mn}x_m=d_n$ ...
0
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0answers
21 views

It would be nice if someone has some idea! (A Diophantine system associated with a network flow)

Assume that we are given a connected network flow with n nodes, $\{1, ..., n\}$, and m arcs. For each arc, say $x_{ij}$ from node i to node j, there is a maximum capacity level given as $M_{ij}$. ...
0
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1answer
28 views

simplifying summations

From knowing $7c=\sum_{i=1}^{50-c}k_i$ and $c\choose 2 $=$ \sum_{i=1}^{50-c}$ $k_i\choose 2 $ how can I get to $\sum_{i=1}^{50-c}(k_i-\mu)^2=(50-c)\mu^2-14c\mu+c^2+6c$ for some arbitrary ...
1
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2answers
46 views

Solve a differential equation systems$\frac{dx}{dt} = -x-6y $ $\frac{dy}{dt} = 3x+5y$

$\mathbf {Consider \space the \space system}$ $$\frac{dx}{dt} = -x-6y $$ $$\frac{dy}{dt} = 3x+5y$$ $\mathbf {Find\space the\space general\space solution\space of\space the \space system.}$ For this ...
1
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2answers
39 views

The number of solutions to a system of equations

Find the value of $k$ for which the system of equations $x-3y-z=0 $ $3x-5y-z=0$ $-x+ky+2z=k^2-4$ has: (i) no real solutions (ii) infinitely many solutions (iii) ...
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1answer
25 views

4th order Runge kutta with system of coupled 2nd order ode MATLAB

I tried using Runge-Kutta methods to approximate motion equations in matlab but it turn out wrong. $$ M\left(\frac{d^2x}{dt^2}\right)=F_n(\cosΦ - u\sinΦ) \\ M\left(\frac{d^2z}{dt^2}\right)=F_n(\sinΦ + ...
0
votes
1answer
37 views

solving an equation $x^x= c$ [duplicate]

I would like to find a solution $x$ for $x^x = c$ where $c$ is a positive constant. Firstly I'm looking for an approximative solution when $c$ tends to infinity. Thank you in advance
1
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0answers
31 views

Systems of second order differential equation

i'm following a course in Hamiltonian systems and regarding the part of linear systems I found this exercise from a book and need to solve it. My ideas are just after the test of the exercise. ...
0
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2answers
48 views

Solving a system of two trigonometric equations

I have to solve the following system made of two equations. The variables are $x_i$ for $i=1,...,n$. For the parameters, we have $a_i\in\mathbb{R}$ and $B\geq0$. The two equations are: ...
1
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1answer
33 views

Linear system 2 unknowns

There are $x$ white and $y$ black pearls and their ratio is $z$. If I add six black and six white pearles, the ratio doubles. I did the following: $ \frac{x+6}{y+6} = \frac{2x}{y}$ and then I get ...
0
votes
1answer
26 views

Are inconsistent equations linearly independent or linearly dependent?

I have a doubt... I know that the systems of equations: $$\begin{cases} x+y=4\\ 2x+2y=8 \end{cases}$$ is LD, and: $$\begin{cases} 3x=4\\ 2y=5 \end{cases}$$ is LI, but what if I have? ...
0
votes
1answer
14 views

System of congrences

If $m$ is an odd integer and $n \in \mathbb N$, prove that the system of congruence $2x \equiv 2n (mod\, m)$ $x \equiv m(mod \, 2^n) $ has exactly one integer solution $x$ with $0 \le x \lt 2^nm$ ...
5
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1answer
144 views

Solution for sets of $(x_k-x_l)(x_l-x_m)(x_m-x_k)>0$

Given a set of inequalities like the following: $$ (x_k-x_l)(x_l-x_m)(x_m-x_k)>0, $$ with $x_n\in\mathbb N_0$. These inequalities have solutions, when $\{x_k,x_l,x_m\}$ obeys a cyclic ordering ...
0
votes
1answer
19 views

How to transform a coupled differential equation into a system with diagonal linear part

Consider the system given by $$iu_t +u_{xx}+2|u|^2u = -v+iu$$ $$iv_t +u_{xx}+2|v|^2v = -u-iu$$ I am trying to transform the system into a system with diagonal linear part. I can solve a problem like ...
1
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0answers
21 views

How to sole this for n? [duplicate]

I started with a hard and unsuccessful way by putting the coefficcients into the matix. I need to solve the system of equation for any natural number n. system
1
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2answers
59 views

How to solve for any given natural number n?

I started with hard way of putting the coefficients into a matrix. But, iz did not help. the following system of linear equations: system
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1answer
95 views

Matrix equation with transpose [closed]

How can I solve this matrix equation for $X$: $$ (A^T)X = B(X-Y)C, $$ where $A^T$ is the transpose of $A$. Here, all matrices are small (e.g., $2\times2$). I am especially interested in the following ...
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votes
1answer
31 views

Monetary question. [closed]

You have 120 papers of dollars. The 10 dollar bills are 10 times the 5 dollar bills, and the others are 100 dollar bills. How many dollars you have ? I tried to set vriables for the 10 dollar bills ...
0
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2answers
65 views

Finding all left inverses of a matrix

I have to find all left inverses of a matrix $A = \begin{bmatrix} 2&-1 \\ 5 & 3\\ -2& 1 \end{bmatrix}$ I created a matrix to the left of $A$, $\begin{bmatrix} a &b &c \\ ...
0
votes
1answer
26 views

Determine the number of solutions of a polynomial system

I have a system of polynomial equations, whose polynomials are all multi linear with coefficients equal to 1. I want to understand if the system has or not real solutions, and if yes, if they are a ...
0
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2answers
41 views

Prove that: $\begin{cases} f(\theta_1)\cos \theta_1=f(\theta_2)\cos \theta_2 \\ f(\theta_1)\sin \theta_1=f(\theta_2)\sin \theta_2 \end{cases}$

Let $f(\theta)$ be a continue function for $\theta\in[\theta_1,\theta_2]$. Prove that: \begin{cases} f(\theta_1)\cos \theta_1=f(\theta_2)\cos \theta_2 \\ f(\theta_1)\sin \theta_1=f(\theta_2)\sin ...
0
votes
3answers
27 views

Necessary condition for uniqueness solution in a system of non-linear equations

Consider a system of non-linear equations with $n$ equations and $m$ unknowns. Is $m=n$ a necessary condition for having one unique solution?
0
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1answer
23 views

proof of existence of solution to componentwise inequality using only linear algebra

For $A \in \mathrm{Lin}(\mathbb{R}^n, \mathbb{R}^m)$ , $m=n$ and $A$ is invertible, I am able to prove the existence of a solution $x$ such that $A x \succeq 0$ where $\succeq$ denotes componentwise ...
0
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0answers
28 views

Which of the following is true about $m \times n $ mayrix of rank $n$.

Let $A $ be an $m \times n$ matrix of rank $n$ with real entries. Choose the correct statement. 1.$Ax=b$ has a solution for any $b$. 2.$Ax=0$ does not have a solution. 3.if $Ax=b$ has a solution ...
1
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0answers
40 views

Closed-form solution for a matrix equation involving pseudo-inverses and Frobenius norms

Let $A\in \mathbb{C}^{m \times n}$ be a wide unknown complex matrix ($m<n$), and $\Sigma\in \mathbb{R}^{p \times n}$ a known rectangular diagonal matrix. $A$ has full row-rank. Is it possible to ...
1
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2answers
29 views

Split complex system of equations into two real systems

Suppose I have a complex system of equations in 3 unknowns, like this one: $$ \pmatrix{ 40 & -20 & 0\\ -20 & 20-20j & 30+10j\\ 4 & -5 & 1 } \pmatrix{ x_1+j x_2\\ y_1+j y_2\\ ...
0
votes
1answer
115 views

Solutions to simultaneous Diophantine equations $2y^2-3x^2=-1$ and $z^2-2y^2= -1$

I am looking for integer solutions for the following set of equations: $2y^2-3x^2=-1$ $z^2-2y^2= -1$ I know that there are the solutions (1,1,1) and (-1,-1,-1) for this set of ...
0
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0answers
33 views

Implementation of Poincaré–Miranda theorem

To test whether a continuous function has one simple root in a given interval $[x0, x1]$ is relatively easy: according to Intermediate value theorem when the sign of function value at $x0$ is opposite ...
0
votes
1answer
37 views

How this type of equation is solved? [closed]

I'm solving a relative and ends when the function (x²+y²)²-4x² derive out this equation, but not that I have to do to get resolve it $$f'x = 4x³+4y²x-8x$$ $$f'y = (4x²+4y⁴)y$$ How solve this ...
3
votes
2answers
61 views

How to solve simultaneous exponential equations with polynomial parts?

I have been puzzled at how to simultaneously solve the following equations and before I give up entirely I thought I'd turn to fellow mathematicians first: $$2^x=3y$$ $$2^y=5x$$ I have graphed both ...
0
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0answers
30 views

Algebraic and geometric representation of the linear system

$1)$ Show that for $\forall\alpha\in\mathbb R$ the set of solutions of a system $$x-y+2z-t=1$$ $$2x-3y-z+t=-1$$ $$x+(\alpha-4)z=\alpha-3$$ is not empty. $2)$ Describe that set for all values $\alpha$ ...
0
votes
1answer
27 views

Determine $a,b\in\mathbb R$ such that for linear transformation $f:\mathbb R^3\rightarrow \mathbb R^3$ is valid: $(4,3,4)\in Im(f)$.

Determine $a,b\in\mathbb R$ such that for linear transformation $f:\mathbb R^3\rightarrow \mathbb R^3$ given by matrix $ \begin{bmatrix} a & 1 & 1 \\ 1 & b & 1 ...
0
votes
3answers
35 views

System of Linear Equations with integer Coefficients

Consider the following system of linear equation: \begin{align} 2a + 4b &= a + 3c\\ 2a + 3b &= 4a + 2b\\ 4a + 2b &= b + nc \end{align} for $a,b,c \in \mathbf{R}_{+}$. How do I ...
1
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5answers
53 views

Four statements, One statement is false math problem

When trying to recall some facts about the ages of his three aunts, Josh made the following claims: Alice is fifteen years younger than twice Catherine’s age. Beatrice is twelve years older than ...
2
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4answers
63 views

Solve for $x$ if $4^{\frac{x}{y} + \frac{y}{x}}$ $=$ $32$ and $\log_3(x+y)+\log_3(x-y)=1$

Question: Solve for $x$ if $4^{\frac{x}{y} + \frac{y}{x}}$ $= 32$ and $\log_3(x+y)+\log_3(x-y)=1$ My attempt: With the first equation $$4^{\frac{x}{y} + \frac{y}{x}} = 32$$ ...
0
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0answers
40 views

How to Calculate Uncertainty in Simultaneous Equations

I need to solve the following simultaneous equation in three variables. I am able to do this, but I am unsure as to what the uncertainty in the values are. a + b = 42.6 ± 0.1 a + c = 56.3 ± 0.1 b + ...
0
votes
1answer
28 views

How to write a transfer function (in Laplace domain) from a set of linear differential equations?

Provided I have a system of linear differential equations (in time domain) such as: $$\begin{cases} \dot{x}(t)=Ax(t)+By(t)+Cz(t)\\ \dot{y}(t)=A'x(t)+B'y(t)+C'z(t)\\ \dot{r}(t)=B''y(t)\\ \end{cases}$$ ...
1
vote
1answer
67 views

Solve the system of equations $\begin{cases}x^3-3x=y \\ y^3-3y=z \\ z^3-3z=x \end{cases}$

Find the number of real solutions to the system of equations $$\begin{cases}x^3-3x=y \\ y^3-3y=z \\ z^3-3z=x \end{cases}$$ Let $f(x) = x^3-3x$ then for $x\in \mathbb{R}-(-2,2)$ we have $x_1 ...
0
votes
1answer
39 views

System of modular equations with unknown modulus

I have a linear congruential generator, which works based on this equation: $X_i = (aX_{i-1}+b) \mod m$. I'm trying to compute next number, but everything I have is given output. So for example $\;a ...
1
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2answers
39 views

How to solve ${(1 + x)^2}y'' + (1 + x)y' + y = 4\sin \ln (1{\text{ + }}x)$?

I don't know how to solve this equation: ${(1 + x)^2}y'' + (1 + x)y' + y = 4\sin \ln (1{\text{ + }}x)$ When it's homogeneous, I try to solve by power series tediously. Is there any good way to ...
1
vote
2answers
39 views

How to find solutions for $a$ and $b$ where $9 \equiv 4a+b \pmod {26} $ and $10 \equiv 19a+b \pmod {26}$? [closed]

$$9 \equiv 4a+b \pmod {26}$$ $$10 \equiv 19a+b \pmod {26}$$ How can I solve the following system?
3
votes
3answers
61 views

Solving system of three quadratic equations

$$\begin{cases} x^2 = yz + 1 \\ y^2 = xz + 2 \\ z^2 = xy + 4 \end{cases} $$ How to solve above system of equations in real numbers? I have multiplied all the equations by 2 and added them, then got ...
0
votes
1answer
17 views

Simultaneous equations with a parameter

Show that the following system of equations has a solution for any value of the constant $\lambda$, using matrix method. \begin{cases} & x+2y+4z = 4 \\& 2x+3y+6z = 0 \\& ...
-2
votes
1answer
47 views

solving a system of simultaneous equations with no values

I need to get the conditions on $a_i$ for the set: $$x_1+x_2+x_3=a_1a_2a_3$$ $$x_1+x_3+x_4=a_1a_3a_4$$ $$x_1+x_2+x_4=a_1a_2a_4$$ $$x_2+x_3+x_4=a_2a_3a_4$$ The best I get is ...