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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30 views

System of differential equations with references to each other

For system of differential equation as follows:\begin{align} \frac{\partial}{\partial t} \begin{pmatrix}\rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{00}\end{pmatrix} &= -\tau i ...
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2answers
24 views

Get variables with Matrix

I try to get the variables for this equation: $$\begin{cases} 6x_1 + 4x_2 + 8x_3 + 17x_4 &= -20\\ 3x_1 + 2x_2 + 5x_3 + 8x_4 &= -8\\ 3x_1 + 2x_2 + 7x_3 + 7x_4 &= -4\\ 0x_1 + 0x_2 + 2x_3 ...
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0answers
47 views

How to solve this system of inhomogeneous differential equations

In some past exam papers for the Maths course that I attend,I found this example and I would really appreciate if someone looked at my solution. It goes like this: Find general solution to $$ y_1' = ...
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1answer
45 views

Solving system of 3 equations

How do I solve the following system? $$ \left\{ \begin{array}{} x_o = 4 - x_r \\ x_r = -2 - x_s \\ x_s = 2 - x_r \end{array} \right. $$ All the techniques i've found for solving 3-equation ...
2
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2answers
65 views

Solve a system of linear equations

$\newcommand{\Sp}{\phantom{0}}$There is a system of linear equations: \begin{alignat*}{4} &x - &&y - 2&&z = &&1, \\ 2&x + 3&&y - &&z =-&&2. ...
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2answers
105 views

How to prove that the level sets of this function are closed curves in a specific region.

I need to study the Hamiltonian differential system $$ \begin{align} \dot{x} &= -2ye^{-x^2}\\ \dot{y} &= 2xe^{-x^2}(1-y^2) \end{align}$$ with Hamiltonian function $$ \begin{align} H ...
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0answers
37 views

Terminology: Name for general, non-differential equations over $\mathbb{R}^n$

Let $x \in \mathbb{R}^n$ be a variable, and $f_1, \ldots , f_n : \mathbb{R}^n \mapsto \mathbb{R}$ be a sequence of functions, each having a closed form expression. Assume that we want to solve the ...
2
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1answer
60 views

Mathematically choose the better discount

This may seem like a homework problem because it is. However it is not my homework - it belongs to the child I am tutoring, so please feel free to give a full answer, as I will only lead the child ...
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3answers
91 views

Solve for $x, y, z$, and $t$

Given the equations: $$6x^2+y^2=z^2\\x^2+ 6y^2=t^2$$ for all $x,y,z,t \in\mathbb{N}$. Solve for $x, y, z$, and $t$.
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2answers
37 views

Method for Finding Matrix-Inverse Through Gauss-Jordan?

When trying to find the inverse of the n$\times$n matrix $A$, one way of going about it is by solving $AX=I$, wherein $I$ is the n$\times$n identity matrix, and $X$ is some n$\times$n matrix which is ...
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3answers
31 views

Question about solving systems of equations (Highschool level)

If I am asked to solve a systems of equation, how would I know which method (substitution, or elimination) to use? What set of conditions should I be looking for, or is it that either method should in ...
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1answer
24 views

Question about solving systems of equations

Is their a universal method to solve systems of equation, eg do methods such as 'elimination' work for ALL types of simultaneous equations (I am specifically referring to 2 and 3 equation simultaneous ...
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2answers
50 views

Euler method application: step size

Suppose we have a system of ODE's: $a' = -a - 2b$ and $b' = 2a-b$ with initial conditions $a(0)=1$ and $b(0)=-1$. How can we find the maximum value of the step size such that the norm a solution of ...
1
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2answers
33 views

System of differential equation with alternative variables

Let us suppose a system of linear differential equations $ \begin{align} \frac{d}{d t}x_1(t)&= -\lambda x_2 (t) \\ \frac{d}{d t}x_2(t)&= \lambda x_1 (t) \end{align} $ How this system could ...
2
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1answer
59 views

Stability of zero

Determine stability of zero in \begin{cases} x'=y \\ y'=-f(x) \end{cases} Here $f: \mathbb{R} \rightarrow \mathbb{R}$ is class $\mathcal{C}^1, f(0)=0$ and $xf(x)>0$ for $x \neq 0$. Could you help ...
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0answers
29 views

Reflection Symmetry for Non-Linear Differential Equations

We are given the equations: \begin{align} \dot{x}& =\mu \, x +y+y^3 \\ \dot{y}& =2x-2y+xy^2+\gamma \, x^2y \end{align} The question at hand is to determine whether there is some sort of ...
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2answers
48 views

Prove that system of equation implies statement

How to prove that $$ \begin{cases} x_1 + x_2 + x_3 = 0 \\ x_1x_2 + x_2x_3 + x_3x_1 = p \\ x_1x_2x_3 = -q \\ x_1 = 1/x_2 + 1/x_3 \end{cases} $$ implies $$ q^3 + pq + q = 0 $$ ?
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2answers
63 views

In search of periodic solutions of a system of ODEs by means of Fourier series

Consider the following non-linear system of ODEs : \begin{cases} x' = y \\ y' = x^2-\lambda x. \end{cases} In search of a solution such that $y(0) = y(2 \pi) = 0$, I am being told to seek $x$ and $y$ ...
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0answers
59 views

Methods of characteristic for system of first order linear hyperbolic partial differential equations: reference and examples

I would like to understand a few points on the methods of characteristics used to resolve a system of coupled, linear first order partial differential equation (of the hyperbolic type). Some example ...
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1answer
17 views

Matrices in systems of linear equations

I've been working on matrices lately. Currently, I am stuck on solving systems of linear equations using matrices. I've read the following article which has proved very helpful in understanding the ...
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1answer
42 views

Are there standard approaches, to solving a system of nonlinear PDE?

If we have a system of PDE's, where each PDE is different i.e. for $u:U\subset \Bbb R^2\to\Bbb R^3$, $u(x,y)=(a(x,y),b(x,y),c(x,y))$, which needs to satisfy $ \left\{ \begin{array}{ll} ...
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0answers
36 views

Solving a set of equations

I have a set of n equations of the form of $Z_i((a-x_i)^2 + (b-y_i)^2 + c^2)$, i varies from 1 to n and $Z_i$, $x_i$, $y_i$ are knowns and $a,b,c$ are unknown. They are all equal to each other ...
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2answers
76 views

How to solve this system of equations that appears in a ODE exercise?

I am trying to solve this equation, we know $A, B, Q,\phi\in\mathbb{R}$. \begin{eqnarray} T''(x) &=& \phi (T(x)-Q) \\ T(0)&=& A\\ T(b)&=&B \end{eqnarray} So the ...
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2answers
34 views

How to solve this equation with three variables with an unknown parameter using Gaussian elimination?

If I've got three equations: $$\begin{array}{ccccccc} x & + & y & + & z & = 3 \\ 2x & + & ay & - & 2z & = 4 \\ x & + & 2y & - & az & = 1 ...
1
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1answer
72 views

Eigenvalues and Eigenvectors for matrix. Complex Eigenvalues

How can I find out the eigenvectors for this matrix: $$A= \begin{pmatrix} -3 &0&0\\ 0&3&-2\\ 0&1&1 \end{pmatrix} $$ I found the eigenvalues: $\lambda_{1}=-3$, ...
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2answers
47 views

solving a system of equations dealing with Lorentz transformations

Can anyone help me to find the solutions of this system of equations: $$c^2x^2-v^2y^2=c^2$$ $$y^2-c^2z^2=1$$ $$vy^2+c^2zx=0$$ I know the answer: $$x= \frac{1}{ \sqrt{1- \frac{ v^{2} }{ c^{2} } } } $$ ...
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1answer
35 views

Find the height of the dam given angles of a triangle

The top of a dam has an angle of elevation of 1.3 radians from a point on a river. Measuring the angle of elevation to the top of the dam from a point 155 feet farther downriver is 0.8 radians; assume ...
2
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3answers
64 views

Why does an $n \times n$ rotation matrix have $\frac{1}{2}n(n-1)$ undetermined parameters?

Consider an orthogonal transformation between Cartesian coordinate systems in $n$-dimensional space. The $n \times n$ rotation matrix $$R = \left(a_{ij}\right)$$ has $n^2$ entries. These are not ...
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3answers
32 views

simultaneous equations with irrational variables

Solve the simultaneous equations $a\sqrt a+b\sqrt b=183$ and $a\sqrt b+b\sqrt a=182$ I made an attempt in vain to equate the coefficients and eliminate
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1answer
31 views

How to solve this system of the 1st order equations?

This is a problem from the book: $$x_1' = x_2\\ x_2' = -x_1\\ x_1(0) = 2\\ x_0(0) = 0$$ The problem says transform the system of the 1st order differential equations into a single differential ...
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1answer
41 views

how to find matrix A from complete solution to Ax=b

I am trying to solve a problem. I was stuck.Any help is appreciated. The complete solution to $Ax=\left[\begin{array}[c]{rr}1 \\3 \end{array}\right]$ is $ x= ...
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1answer
48 views

For this 2 by 2 locally linear system, how to determine that this “indeterminate” critical point is a centre? Boyce, p516, Question 9.3.12

$12.$ (a) Determine all critical points of $\dfrac{dx}{dt}=(1+x)\sin y$ , $\dfrac{dy}{dt}=1−x−\cos y$ . (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of ...
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1answer
351 views

The solutions for the equation $\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$

How can I find the solution for the following equation in $a,b \mbox{ and } c$. $$\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$$ Also $b-c \neq 1$, $c-a \neq 1$ and $a-b \neq 1$. Thanks!
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0answers
44 views

Strong Lyapunov Function

By showing that $V(x_1,x_2) = (x_1)^2 + (x_2)^2$ is a strong lyapunov function for the system: $x_1’ = -x_2$ $x_2’ = x_1 + (x_2)^3 - x_2$ determine a region of ''attraction'' for the origin. I ...
0
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2answers
49 views

Solve Linear Sytem of Equation for $u,v,w$

I need to solve this sytem for $u,v,w$. I´ve tried basic algebra, but my answer does not mach the one from the book.
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0answers
46 views

Solving the equation $AX+XA' = 0$

I am trying to solve the equation $AX + XA' = 0$ I could find how to solve when "$+$" is a "$-$" or $X$ is conjugated instead of $A$. Is there a solution for this problem too? In particular, I am ...
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2answers
21 views

Solve system of two conic equation

I need to solve three types of system of equations in general form: System of two linear equation ($Ax + By + C = 0$) which can be done perfectly by calculating D, Dx, Dy. System of two equations ...
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1answer
35 views

Show that the solution of the differential system are periodic.

Let $y,z$ two functions defined on $\mathbb{R}$. Show that the solution of the differential system : $$ y'=z^3 \qquad z'=-y^3 $$ are periodic. My attempt : With some works I can show that ...
2
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1answer
56 views

Is it possible for a system of equations to have a non-zero determinant and no solution at the same time?

I am quite confused by the solution I was given for the following problems: a) Solve the following system of equations using Gauss elimination only: $2x - y = 5$ $-x + 2y = -4$ $3x - y = -1$ b) ...
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1answer
63 views

Solve the recurrence relation $f(n) = f(n - 1) + f(n - 2) + f(n - 3)$

This is a problem I was playing with that troubled me greatly. $f(n) = f(n - 1) + f(n - 2) + f(n - 3)$ $f(1) = f(2) = 1$ $f(3) = 2$ So, the goal is to try and find a solution for f(n). I tried ...
2
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1answer
70 views

Method of characteristics for a system of pdes

I can do parts a) and b) as follows $\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1\end{pmatrix}\frac{\partial}{\partial{}x}\begin{pmatrix} u \\ v \\ w\end{pmatrix}+\begin{pmatrix} ...
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1answer
40 views

PRobability Markov chain, system of equations

I'm looking for techniques or tricks to solve a system of linear equations you get where you want to find the limiting probabilities. The system is this: $\pi_0 = 0.7\pi_0 + 0.2\pi_1 + 0.1\pi_2$ ( ...
4
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2answers
57 views

Solve numerical system of nonlinear equations?

I need to solve a nonlinear system of equations that looks like this ...
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1answer
51 views

System of equations with a unique solution, no solution or an infinite number of solutions

I was doing a past OCR Further Pure 1 Paper from January 2011, but came across the following question that I could not solve, even with the help of the mark scheme: Determine whether the ...
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0answers
31 views

a system of linear equations $x-y+z=0$

Yall are probably gonna think me a noob. But I am working on this eigenvector problem and I reduced the matrix to $x-y+z=0$ . How do I describe this solution set. I know how to do it if it's just ...
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1answer
31 views

Using Laplace Transform to solve a 3 by 3 system of differential equations

I have been trying to solve this system of equations using Laplace transforms for a while. It is very easy to solve it using eigenvalues and eigenvectors, but when I tried to do it using Laplace I ...
0
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0answers
10 views

Finding an isometry in high dimension

Given two points $x,y \in \mathbb{R}^D$, I am looking for the isometry $A\in \mathbb{R}^{D\times D}$ that maps $x$ to $y$: $Ax=y$ I want to determine the $D^2$ parameters constituting $A$. When is ...
0
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1answer
40 views

Finding equations when given new center of a circle

$y = −x + \sqrt{2}$, $y = −x − \sqrt{2}$, $y = x + \sqrt{2}$, and $y = x − \sqrt{2}$. These equations determine lines, which in turn bound a diamond shaped region in the plane. Construct a diamond ...
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2answers
121 views

A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$

Let: $f(x)$ and $g(x)$ be twice differentiable, non-decreasing functions. $f''(x) = g(x)$ and $g''(x) = f(x)$. $f(x) \cdot g(x)$ is a linear function. Then we have to show that $f(x) = g(x) = ...
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3answers
73 views

How do we know that if $Ax = b$ has a unique solution, $A$ is invertible?

We are of course assuming $A$ is an $n\times n$ matrix. I know there's a proof of it going the other way (invertibility implies a unique solution), but I'm trying to work out a proof going this way. ...