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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20 views

Solving system of equations in a 3$\times$2 matrix

I am trying to the following system of equations: $qx-y = q-2$ $x-py = 2-p$ $2x-y =0$ However, I am confused as to how a $ 3 \times 2 $ matrix should look like after doing the row echelon form of it. ...
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2answers
40 views

If $\displaystyle \frac{x}{a}+\frac{y}{b}=1$ and $\displaystyle \frac{a^3}{x}-\frac{b^3}{y}=b^2-a^2$, prove that $x^2-y^2=a^2-b^2$

The question, taken from a math book for competitions, goes as follows: Take the system $\displaystyle(\Sigma)=\begin{equation} \begin{cases} \displaystyle\frac{x}{a}+\frac{y}{b}=1\\ \...
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0answers
13 views

Prove that the obtained system of equations is nonlinear [closed]

I am trying to prove that the next system is nonlinear. I know the unknown functions, can someone lead me to the answer?
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1answer
22 views

Can there be more than one value for scalars between lines that intersect?

Determine the intersection point of the line through the points (1,−2,13) and (2,0,−5) and the line given by r(t)=⟨2+4t,−1−t,3⟩ or show that they do not intersect. I simply treated each point as a ...
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34 views

Resolution of this system of equation

I am currently on the verge of a nervous breakdown for an apparent easy-to-solve system of equations. I may miss something to get the key of the problem if the system is solvable but it should be. Let ...
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1answer
21 views

Solving a system of nonlinear equations with derivatives inside (for spin 1/2 particle in magnetic field)

I have been trying to solve the state vector for a spin 1/2 particle inside an oscillating magnetic field, and having managed with the field in the z axis, I tried it instead in the x-axis. This ended ...
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1answer
50 views

Solving for matrix X (using the Moore-Penrose Pseudoinverse)

Consider the following matrix equation: $\mathbf{B'XB=A}$ where $\mathbf{X}$ is $m \times m$ $\mathbf{B}$ is $m \times n$ $\mathbf{A}$ is $n \times n$ $\mathbf{A}$ and $\mathbf{B}$ are known $\mathbf{...
2
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1answer
26 views

Euclidean norm of a solution of matrix differential equation

I try to solve a problem I have found in a book where I'm asked to find the solution to a differential equation of the form $$ x'=Ax $$ where $A$ is a matrix. The answer is the exponential of $e^{At}$....
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2answers
54 views

Is there a method for deriving a function given a system of equations containing references to said function?

For example, there's the archetypical Fibonacci sequence, which can be defined as $$ \begin{align} F(0) &= 0\\ F(1) &= 1\\ F(n) &= F(n-1) + F(n-2). \end{align} $$ The only continuous ...
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58 views

Approximate solution of a non-linear system

Is there a method to find an approximate solution of the following system of nonlinear equations, with this type of exponents? \begin{equation} \left\{\begin{matrix} \dfrac{x^{20}-1}{x-1}+\dfrac{z^{...
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1answer
20 views

Solving this system for Partial Fraction Decomposition

I have a rational fraction $\frac{P(x)}{Q(x)}$ and would transform it into a sum of separate fractions. I know that $\{a_n\}$ is the set of the roots of $Q(x)$ which is of grade $t$, so it has exactly ...
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0answers
28 views

Numerical divergence of some solution of the restricred three body problem

In this question on the restricted three body problem are described very well the mechanics of the following system of ODEs: \begin{cases} y_1'' = y_1 + 2y_2' - \mu_2 \frac{y_1+\mu_1}{D_1} - \mu_1 \...
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1answer
46 views

Solving two equations with four variables over integers

Find all integer tuples $(a,b,c,d)$ such that $$ab-2cd-3=ac+bd-1=0.$$ The substitution $ab=p,cd=q,ac=r$ and $bd=s$, leads to a quadratic in $r$ (messy algebra) $$r^2-r+(3q+2q^2)=0$$ Since the ...
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0answers
50 views

How to solve none symmetry system of equations in Real numbers?

I am trying to solve this system of equations in the set of all real numbers. Solve the system of equations $$\begin{cases} x^3+y (y-z)^2=2, \\ y^3+z(z-x)^2=3, \\ z^3+x(x-y)^2=8. \end{cases}$$ I ...
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3answers
66 views

If the systems $Ax=u$ and $Ax=v$ don't have solutions, is it possible that $Ax=u+v$ has a solution?

For instance, for $u=(0,2,4,6)^T$ and $v=(1,3,5,7)^T$, is it possible that the system $$ Ax=u+v $$ has a solution if $Ax=u$ and $Ax=v$ have no solution? I know that if $Ax=b$ and $Ay=b$ with $x$ and $...
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0answers
36 views

Prove that a system $Ax>b$ (i.e. $(Ax)_i>b_i$ for each $i$) is solvable iff $uA=0,u\geq0,u\ne0$ imply $ub<0.$

Prove that a system $Ax>b$ (i.e. $(Ax)_i>b_i$ for each $i$) is solvable iff $uA=0,u\geq0,u\ne0$ imply $ub<0.$ I've managed to show that $(\Rightarrow)$ holds but don't know what to do with ...
3
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2answers
104 views

Can $x^2+y$ and $y^2+x$ both be perfect squares for $x,y$ positive integers?

I was recently idly reading some of the problems of the All Soviet Union Math Competitions, when I came across a fascinating problem from the 1966 edition, specifically problem 3: Can $x^2+y$ and $y^...
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0answers
7 views

Maximum time of existence of the solution of the system of differential inequalities

I'm need to check if the maximum time of existence of the solution of the system of differential inequalities: $$\left\{\begin{matrix} y'(t)\geq y^p - C_1\\ y'(t) \geq z^q-C_2 \end{matrix}\right. ...
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0answers
28 views

Period 1 Fixed Points of a difference equation

So i want to determine all the period 1 fixed points and analyse its stability in terms of the parameters v and c $$\begin{cases} x_{n+1}=ax_{n}+by_{n} \hspace{0.1cm}\\ y_{n+1}=-bx_{n}+ay_{n}-8\pi c \...
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1answer
44 views

Is it possible to solve the following 3 equations using matrices?

Is it possible to solve the following 3 equations using matrices? $x^2 + y^2 = r^2$ $(x-2)^2 + (y-6)^2 = r^2$ $(x-8)^2 + (y-4)^2 = r^2$ I am attempting to find the equation of a circle given 3 ...
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0answers
32 views

How to find a continuum of solutions to forward equations in 2 variables [closed]

I have a system of equations in two variables $(x_t,y_t)$ given by $x_{t+1}=f(x_t,y_t)$ and $y_{t+1}=g(x_t,y_t)$. The continuous function $g$ is such that $0\leq y_{t+1}<y_t$. The continuous ...
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0answers
16 views

repeated eigenvalues, 5×5 matrix with 1 eigenvalue of multiplicity 5 and 3 eigenvectors.

If 5×5 matrix A has an eigenvalue of multiplicity 5 with 3 corresponding eigenvectors, how do I go about finding the other 2 solutions of $y^\prime=Ay$? If there was only one eigenvector $\mathbf{k}$, ...
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1answer
29 views

write new boundary conditions to a system of ODEs

Suppose we have the following system of equations: \begin{cases} \ddot x_1 = f(x_1, x_2) \\ \ddot x_2 = g(x_1, x_2) \\ x_1(0) = \alpha \\ x_2(0) = \beta \\ \dot x_1(0) = \gamma \\ \dot x_2(0) = \delta ...
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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(...
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0answers
25 views

Gaussian elimination: all variables in row are zero but constant is non-zero?

I'm starting linear algebra next week and been trying to get a little ahead. I'm working with a practice sheet I found through Google. One of the problems has the equations $2x+5y=9$ $x+2y-z=3$ $-3x-...
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2answers
68 views

Simultaneous equations using distances in polar coordinates

Working on a project and this problem came up. Trying to find a general form for $r_q$ and $\theta _q$ from these two simultanous equations. I'm think that there will be many cases (almost all of them)...
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0answers
54 views

How to solve the coupled system of ODES $\dot{x}=y^2$, $\dot{y}=x$?

How to solve the following system of coupled ODEs? $$ \begin{aligned} \dot{x} &= y^2 \\ \dot{y} &= x \end{aligned} $$ I can see how to solve for their trajectories as $dy/dx$ is of course ...
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0answers
65 views

Partial derivative of function evaluated at fixed point

Consider a function $f\left(x,y,\Lambda\right)$ where $\frac{\partial f}{\partial x}<0$, $\frac{\partial f}{\partial y}>0$ and $\frac{\partial f}{\partial \Lambda}>0$. We know that $x>0$, $...
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0answers
21 views

Need help solving system of equations [duplicate]

$\begin{cases} x + y + z=6 \\ x^2 + y^2 + z^2=24 \\ x^3 + y^3 + z^3=96 \end{cases} $ Solutions are supposed to be $(x,y,z)=2, 2+\sqrt6, 2-\sqrt6$ I have found that $xy+xz+yz=6$ and $xyz=-4$ but i dont ...
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1answer
40 views

Solving system of differential equations : Wolfram Alpha vs theorem

I am burning my brain finding the most correct way to solve a system of differential equations. Here is an example : $$\begin{cases} x'=5x-2y\\ y'=-x+6y \end{cases} $$ Let's $Y(t)=\begin{pmatrix} x(t) ...
0
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1answer
70 views

solution to $\sum_{i=1}^{n}\frac{1}{a_{i}x+b_{i}} = 0$

Is there any general procedure to solve the equation $$ \sum_{i=1}^{n}\frac{1}{a_{i}x+b_{i}}=0 $$ with respect to $x$ for given $a_{i}$ and $b_{i}$, with $i=1,\dots,n$?
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1answer
38 views

Does the following system of linear equations contain infinite solutions?

Consider the following system of linear equations $$ \begin{align} x+y+z+w &= 2 \\ x- y+2z+3w &=-10 \\ x -3y+3z+5w &= -7 \\ x+3y- w &= -1 \end{align}$$ If we add the first two (solve ...
0
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1answer
29 views

Reverse a Generic Trimetric Projection [closed]

Trimetric Projected Tile I am creating a grid using Trimetric Projection tiles, and have a way to go from a standard 2D grid's coordinates to a generic Trimetric tile coordinate using the following ...
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0answers
40 views

On a general system of linear equations in a derivative problem

For some days a problem that I encountered in the Calculus textbook of mine has been preying upon my mind. The problem is as follows: Determine the expression for the polynomial $F(x)$ given that it ...
0
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2answers
42 views

Particular solution of a system of second order ODE

I have the following system of two ODEs: $$\begin{bmatrix} -k & k & \\ k & -k & \\ \end{bmatrix}\begin{bmatrix}x_1(t)\\x_2(t)\end{bmatrix}+\begin{bmatrix}F-k\times a\\k\times a\end{...
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0answers
22 views

Solution to a linear equation involving a skew-symmetric tensor

Say that $S(\mathbf{x})$ is a skew-symmetric $k+1$-tensor, that is, $S_{i_0,...,i_a,...,i_b,...,i_{k}}(\mathbf{x})=-S_{i_0,...,i_b,...,i_a,...,i_{k}}(\mathbf{x})$ for $a,b=0,...,k$, then find $S(\...
2
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1answer
116 views

Weak shock solutions of system of conservation laws

We consider the system of autonomous conservation laws $$\mathbf{u_t} + (\mathbf{f(u)})_x = 0$$ with left state $\mathbf{u}_L = \mathbf{u}_0$ and right state $\mathbf{u}_R = \mathbf{u}_0 + \epsilon\...
2
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1answer
137 views

Find $x_1 + x_2 + \dots+ x_{n}$ and $1^{n+1}x_1 + 2^{n+1}x_2 + \dots + {n}^{n+1}x_{n}$ given a set of linear constraints

$x_1, x_2, ..., x_{n}$ satisfies $$1^j x_1 + 2^j x_2 + \dots + n^j x_{n} = {(n+1)}^j,$$ where $j = 1, 2, ..., n$. Find $x_1 + x_2 + \dots + x_{n}$ and $1^{n+1}x_1 + 2^{n+1}x_2 + \dots + n^{n+1}x_{n}...
0
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0answers
39 views

Solving a set of equations [duplicate]

Well, I am trying to solve the following system of equations for $\left(\text{a},\text{b},\text{c}\right)$: $$ \begin{align*}\begin{cases} \text{a}+\text{b}+\text{c}&=\alpha_1\\ \\ \text{a}^2+\...
0
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1answer
75 views

Find all solutions to the system of equations $a+b+c=1$, $a^2+b^2+c^2=2$, $a^4+b^4+c^4=3$ [duplicate]

Find all solutions to the system of equations $$a+b+c=1$$ $$a^2+b^2+c^2=2$$ $$a^4+b^4+c^4=3$$ By squaring the first equation and substituting value of $a+b+c$ we get $\sum ab=-\frac{1}{2}$ From ...
5
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1answer
72 views

Solve system of 2 equations with 3 unknowns

We are given a triangle $ABC$ with sides $a, b, c$ respectively and for which the following relationships hold: $a^2+bc\sqrt 3 = b^2+c^2$, $c^2+ba = a^2+b^2$ We want to prove that angle $B$ is right. ...
1
vote
1answer
169 views

Simple system of matrix ODEs

We are given $A \in \mathbb R^{n \times d}, b\in \mathbb R^{n \times 1}$ where $d > n$ and $rank(A) = n$, and initial values $W_0 \in \mathbb R^{d \times d}, x_0 \in \mathbb R^{d \times 1}$ I'm ...
0
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0answers
27 views

Systems | Linear independency

Okay, hello, so I have to prove two propositions. Ok, so here is what I have in my course. "Because any linear operator between $R^n$ and $R^n$ (that are perfectly defined by a matrix of n rows ...
0
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1answer
34 views

Why isolating a vector becomes transposed?

I am following the PCA course from Coursera and while the instructor was isolating the beta coefficient from the following equation; $ x_{n} = \sum_{i=1}^{D} \beta_{in}b_{i} $, he ended with $\beta_{...
0
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0answers
52 views

Rankine-Hugoniot for system of autonomous conservation laws

We have a system of autonomous conservations laws: $$ \boldsymbol{u}_t + (\boldsymbol{f}(\boldsymbol{u}))_x=0 $$ with shock solutions of Riemann problem, left state $ \boldsymbol{u}_L=\boldsymbol{u}_0 ...
2
votes
4answers
91 views

How to solve coupled second order differential equations

I have the following coupled differential equations: $$ 2y''- 3y' + 2z' + 3y + z = e^{2x}$$ $$y''- 3y' + z' + 2y - z = 0 $$ I'm not sure how to solve them as if I try $y = Ae^{\lambda x} $ and $z = Be^...
-1
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0answers
36 views

How can I solve this system of differential equations with eigenvalues? [closed]

I have this system of linear ODEs. $$ {\dot {\bf x}} (t) = \begin{bmatrix} -\sqrt{7} & 0 \\ 0 & -\sqrt{7} \end{bmatrix} {\bf x} (t) $$ How can I solve this system of differential equations ...
1
vote
2answers
29 views

Help to find the solution of $y' = \begin{pmatrix}-2 & 1 \\1 &-2\end{pmatrix} \cdot y + \begin{pmatrix}2\sin(t) \\ 2(\cos(t) - \sin(t))\end{pmatrix}$

I'm currently studying a course on numerical solutions of ODEs and I've been given this system of differential equations: $$y' = \begin{pmatrix}-2 & 1 \\ 1 &-2\end{pmatrix} \cdot y + \begin{...
0
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0answers
58 views

Solutions to a linear system

My question is can the pairs $$(1, 2), (4, 8), \text{and } (−1, 5)$$ all be solutions to a linear system with two equations in two variable? My intuition says no as the pairs $(1, 2), (4, 8)$ are ...
0
votes
0answers
37 views

Comparison Theorem for a system of ODEs

Based on the comparison Theorem we know for continuous $u,v$ and diffrentiable on $[a,b]$ and continuous map $f$: if $u(a)<v(a)$, and $\dot{u}-f(t, u)<\dot{v}-f(t, v)$ $\rightarrow$ then $u<v$...

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