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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31 views

Classifying of differential equations

this is exercise 1.3 and 1.6 from Geralds book (which is free: http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf). The exercises I'm about to present are short and very related (I think), ...
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2answers
60 views

What functions $u(x)$ and $v(x)$ satisfy $u(x) = v'(x)$ and $u'(x) = v(x)$?

I recently came across a problem in a calculus textbook which involved functions satisfying the relation $u(x) = v'(x)$ and $u'(x) = v(x)$. The problem didn't list any specific functions that for ...
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1answer
26 views

Determine whether or not( (x+y+z)==(p+q+r) and E1==E2 and (x!=p or y!=q or z!=r))

there are two number E1 and E2 $$E1=Ax + By + Cz \quad\mbox{and}\quad E2=Ap + Bq + Cr$$ Value of $A,B,C$ are different and positive integers. Value of $x,y,z,p,q,r$ may be same and they are ...
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0answers
8 views

Eliminating dependent variables

consider $Ax=b$ in which $A$ is a symmetric matrix of size $m$ and rank $m-r$. On the other hand, $B_{r\times m}x=0$ provides $r$ equations by using which the system of equations can be reduced to a ...
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1answer
31 views

Trouble manipulating equations

Can someone please help me find the values of $a$, $b$ and $e$? Thank you so much! 1) $\frac{1}{2}e - 4a = 0$ 2) $\frac{1}{2}e - 2b = 0$ 3) $\frac{3}{8}(1 + a) + \frac{1}{8}(1 + b) - 2e = 0$
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2answers
50 views

The only natural number $x$ for which $x+\sqrt{-2}$ is a cube in $\mathbb{Z}[\sqrt{-2}]$ is $x=5$

Let $A = \mathbb{Z}[\sqrt{-2}]= \{a+b\sqrt{-2} \ : a, b \in \mathbb{Z}\}$. Show that the only natural number $x$ for which $x+\sqrt{-2}$ is a cube in $A$ is $x=5$. So I have to show that there ...
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2answers
55 views

$a$,$b$ and $c$ are roots of the equation $x^3-x^2-x-1=0$

The roots of the equation $x^3-x^2-x-1=0$ are $a$,$b$ and $c$. if $n \gt 21 $ and $n \in \mathbb{N}$ The find the possible values of $$E=\frac{a^n-b^n}{a-b}+\frac{b^n-c^n}{b-c}+\frac{c^n-a^n}{c-a}$$ ...
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0answers
11 views

Finding eigenvalues of system of PDEs.

I have the following system of PDEs for $\phi(x,t)$: $$ \phi_{xx} = -\lambda \phi \\ \phi_t = 4\phi_{xxx}, $$ with $\lim_{|x|\to \infty } |\phi(x,t)|= 0$. I need to find the (possibly complex) ...
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1answer
21 views

How to explain some gaps in understanding of the following interpretation of a given proof?

I've been teaching myself linear algebra, and have come to the point where I'm studying the theorem which states that RREF matrices are always unique. I found this proof, but I'm not sure if my ...
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0answers
16 views

Why does biconjugate gradient (BiCG) work for nonsymmetric matrices?

After looking through the derivation of CG, I understand why it requires the coefficient matrix $A$ to be symmetric, since the property is used to produce a short recurrence relation for the ...
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1answer
26 views

Show that two ways to sum the entries of an $n\times n$ array yield the same

Let $A = (a_{ij})$ be a matrix. I am looking for an algebraic proof that $$ \sum_{\sigma, \tau \in S_k} \mbox{sgn}(\sigma\circ\tau) a_{1\sigma(1)}a_{2\sigma(2)}\cdots ...
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2answers
56 views

Geometrical interpretation of solving a 3x3 system of equations

Solve the following system of equations and give a geometrical interpretation of the result. \begin{align*} x + y + z &= 6\\ 2x + y − 3z &= -5\\ 4x − 5y + z &= −3 \end{align*} I know that ...
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3answers
29 views

Find $\frac{y}{x}$ from $3x + 3y = yt = xt + 2.5x$

I need to find the ratio of $$\frac{y}{x}$$ If given that $$3x + 3y = yt = xt + 2.5x$$ So what I tried is: $$t = \frac{3x + 3y}{y}$$ And then put it in the equation $$\frac{x(3x + 3y)}{y} + 2.5x ...
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2answers
33 views

It is known that $y = ax^2 + bx^3$; when $x = 2$, $y = 5.6$, and when $x = 3$, $y = 25$. Find the values of a and b.

$$5.6 = 4a + 8b$$ $$25 = 9a + 27b$$ $$5.6 - 25 = ( 4a + 8b ) - ( 9a - 27b )$$ $$-19.4 = -5a - 19b$$ I'm stuck at that point. $$a = -1.36$$ $$b = 1.38$$
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2answers
32 views

How do I give an algebraic specification for the range of a matrix?

I am given the following $3 \times 3$ matrix: $$\begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 1 \\ 2 & 2 & 10 \end{bmatrix} .$$ Once reduced, I get the following augmented matrix: ...
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2answers
49 views

Resolve $ \frac{120}{x+y} + \frac{60}{x-y} = 6;\,\frac{80}{x+y} + \frac{100}{x-y} = 7$

I want to resolve this system of equations: $$\begin{cases} \frac{120}{x+y} + \frac{60}{x-y} = 6 \\\frac{80}{x+y} + \frac{100}{x-y} = 7\end{cases}$$ I came to equations like $$x - \frac{10x}{x-y} + ...
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0answers
19 views

Is it possible to solve this system of nonhomogeneous DE?

I am following this ARTICLE on solving the differential equations. Take a closer look at page 580, chapter "11.7 Nonhomogeneous Linear Systems" - Theorem 23. Follwing this work, brought me to the ...
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1answer
55 views

Simplifying a complex trigonometric expression

Context: In a previous question , I've stated I'm making a program that will be used for calculating stuff with Statics of a particle. I've come across another scenario in which there's three forces ...
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2answers
131 views

Radius of a largest circle inscribed under $y=\frac{1}{(1+x^2)^n}$, closed form

The curve $y=\frac{1}{1+x^2}$ has an obvious connection to circles, because it's the derivative of the arctangent function. Besides, if we inscribe a circle under it, its radius is exactly ...
2
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1answer
93 views

How to find dispersion relation for a system of linear ODEs

I am trying to find the dispersion relation for a system of linear ODEs. I can do this for a single linear PDE, for example $$u_x = u_t$$ by substituting $u = Ae^{i(kx-wt)}$, here $w = w(k)$ where ...
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2answers
31 views

system of equation with cosh and sinh

is there simple a way solve this system to find the unknown x and y $$cosh\frac{a+x}y=\frac{b}{y}$$ $$sinh\frac{a+x}y=tanθ$$ My attemp: dividing these equations we get ...
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1answer
22 views

Four mutually exhaustive sets; finding the intersection of two sets. Set theory and system of equations.

In a group of 120 persons there are 80 elements of B and 40 elements of G. Further 70 persons in the group are M and the remaining are H. Then the number of elements that are both in B as well as M. ...
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0answers
25 views

System of ODEs: unstable node

For a 2x2 ODE system $\mathbf{x}' = A\mathbf{x}$, where $A$ has distinct positive eigenvalues $0<\lambda_1<\lambda_2$, and $\mathbf{x}(t) = (x_1(t),x_2(t))^T$, the origin in the $x_1 x_2$-plane ...
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3answers
38 views

How can I prove that in a linear system can be there are three solution sets without use matrices?

I would like to know how to prove that in a linear system can be there are three solution sets (empty, a set with one solution or a set with infinitely many solutions) without use matrices? Thanks in ...
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1answer
38 views

Prove $x_2$ is a sol of $x'=Ax$ given $x_1$ is a sol.

Suppose $2 \times 1$ vector $x_1(t) = [u(t), v(t)]^T$ is a solution of $x' = Ax$ where $A$ is a $2 \times 2$ matrix of real numbers. Prove that $2 \times 1$ vector $x_2(t) = [ku(k^2t), ...
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2answers
82 views

Contradictory equations

Question: Find whether the solution set of $$\begin{cases}2x = 1\\ y + 5 = x\\ x = y + 3\end{cases}$$ is a singleton. My attempt: Rewriting the first equation will give us $x = \frac{1}{2}$. The ...
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0answers
17 views

Solving polynomial systems with homotopy. Where is the bottleneck?

I have a polynomial system with $n$ unknwns ($n$ can be between 3 and 20), that is known to have at least $n!$ isolated solutions. I want to solve this system numerically, but if I plug it in an ...
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3answers
65 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 ...
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0answers
13 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 ...
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0answers
74 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 ...
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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 ...
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1answer
37 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) $a_{1n}x_1 + a_{2n}x_2 + \cdots + a_{mn}x_m=d_n$ ...
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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 $M_{ij}$. ...
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1answer
25 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 ...
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2answers
45 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 ...
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2answers
38 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
21 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 + ...
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1answer
36 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
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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. ...
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2answers
47 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: ...
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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 ...
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1answer
24 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? ...
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1answer
13 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 $0 \le x \lt 2^nm$ ...
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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 ...
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1answer
17 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 ...
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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
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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
87 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 ...