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.

learn more… | top users | synonyms (1)

6
votes
0answers
136 views

How to solve a particular initial-boundary value problem

I have the following initial-boundary value problem $$\begin{cases}\dfrac{\partial^2 u_1}{\partial x^2}=A_{11}\dfrac{\partial u_1}{\partial t}+A_{12}\dfrac{\partial u_2}{\partial ...
4
votes
0answers
151 views

When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
4
votes
0answers
43 views

Is there a name for systems of equations with min and max functions included?

In a big project I'm working on, I'm running into systems of equations that look like the following: $$a = \min(b, c)$$ $$b = d^2 + a$$ $$c = \max(a + b, d)$$ Basically, nonlinear systems of ...
4
votes
0answers
95 views

if $(n+1)c_{n}=nc_{n-1}+2nd_{n-1},d_{n}=2c_{n-1}+d_{n-1}$How prove $c_{n},d_{n}$ are integers

let sequences $c_{n},d_{n}$ such $$c_{0}=0,d_{0}=1$$ and for $n\ge 1$,have $$c_{n}=\dfrac{n}{n+1}c_{n-1}+\dfrac{2n}{n+1}d_{n-1}$$ $$d_{n}=2c_{n-1}+d_{n-1}$$ show that $c_{n},d_{n}$ are integers. my ...
4
votes
0answers
44 views

Solving a system of equations

I'm trying to prove the existence of a solution to the system of equations $$c_i = \gamma x_i + (1-\gamma) \frac{x_i^2}{\sum_{j=1}^\infty x_j}$$ for $i\in\{1,2,....\}$ where $\sum c_i=1$. I am also ...
3
votes
0answers
33 views

Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
3
votes
0answers
59 views

Writing ODE system with a complex variable

I'm looking at the system of ODEs: $$\begin{cases}\dot{x} = -y + kx + xy^2\\ \dot{y} = x + ky - x^2\end{cases}$$ I'm trying to introduce a complex variable $z = x+iy$ to write this as a single first ...
3
votes
0answers
33 views

Why are equilibria so important?

In studying nonlinear systems of differential equations, unlike linear systems, it turns out that we are more interested in equilibrium points rather than general solutions themselves. I mean, look ...
3
votes
0answers
41 views

Interpretation of a 3 Variable System of Equations

I'm a high school student, and, of course, this week is finals week. For my Algebra 2 semester final, we have been permitted to take the test home and collaborate with others. This final can be viewed ...
3
votes
0answers
63 views

Existence of Periodic Solution

I'm working with the system of equations below that represents a Pendulum with constant forcing. \begin{align*} \theta'&=v\\ v'&=-bv-\sin(\theta)+k \end{align*} Where $\theta$ gives the ...
3
votes
0answers
42 views

Recovering a kernel from a system of equations

Suppose $f\in C([0,\frac{3}{4}]^2)$ and $$\begin{array}{rlr}\text{i.}& \int_0^{\frac{3}{4}-x} f(x,y)dy=-\frac{1}{2}x^2+\frac{9}{32}&\forall x\in [0,\frac{3}{4}]\\ \text{ii.}& ...
3
votes
0answers
55 views

Sharkovskii's theorem in two dimensions?

Question A weak form of Sharkovskii's theorem in $1$D dynamical systems states that, if a continuous function $f:I\to I$ does not include a periodic point of least period $2$ on $I$, then ...
3
votes
0answers
122 views

Algorithms for solving overdetermined, homogeneous linear systems with multivariate polynomial coefficients

I would like to solve overdetermined, homogeneous linear systems of equations with multivariate polynomial coefficients, i.e., $Ap=0$ with $A$ an $m\times n$ matrix, $m\gg n$, and $a_{i,j} \in ...
3
votes
0answers
37 views

Solving a system of first order differential equations

So, I have (another) problem with differential equations (from an optimal control problem). I am trying to solve the following system of DEs (is this even a system?): $$ \lambda'(t) = r \lambda(t) + ...
3
votes
0answers
47 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
3
votes
0answers
60 views

Solving a system of 3 variables

How to solve or what is the algorithm to solve a system of equations like this: $$\eqalign{ (x +\phantom{3} z)^2 + (y +\phantom{3} w)^2 &= 52\cr (x + 3z)^2 + (y + 3w)^2 &= 296\cr (x ...
3
votes
0answers
81 views

An interesting system of equations

We have the following system with a and b, real numbers: $ax+y + z =4$ $x+2y+3z=6$ $3x-y-2z=b$ Show that $\forall a \in \mathbb{Z} $ there is a $b \in \mathbb{Z}$ such that the system admits a ...
3
votes
0answers
48 views

Qualitative dependence of solution to second-order matrix differential equation on eigenvalues

Suppose we have a matrix differential equation in $\vec{x}(t)=\left(\begin{smallmatrix}x_{1}(t) \\ \vdots \\ x_{n}(t)\end{smallmatrix}\right)$, such that: ...
3
votes
0answers
38 views

Buchberger`s algorithm - Performance?

I want to solve an equation system with 3 complex variables, 3 complex equation and maximum degree 3. Is it reasonable to do this with Buchberger`s algorithm or should I better do it with an ...
2
votes
0answers
17 views

Function intersecting 3 points & deriviate is positive for a range of x values

Thank you for taking the time to help out on this question. I'm looking for a function that intersects 3 points, and a derivative for every value of x between x=0 and x = 365 where dy/dx >= 0. My ...
2
votes
0answers
50 views

How to find whole number answers in systems of square root equations

Given the following 4 equations, can you find 4 whole number answers using whole number variable inputs? $x,y,z$ where $x>y>z$ $Eq 1 = (x^2-2xy+y^2-2xz+z^2)^{\frac{1}{2}} $ $Eq 2 = ...
2
votes
0answers
27 views

How to solve the equation $Au+Bv=C$

How do I solve $Au+Bv=C$ Where $A$ and $B$ are constant known matrices that are nxn, $C$ is a constant known nx1 vector while $u$ and $v$ are unknown nx1 vectors with the condition given that $u_i = ...
2
votes
0answers
21 views

Solve Intergal Equation of form g.u1=Int(K.u2) for u1 and u2

I'm trying to find a solution to a differential equation of an unusual form: $$g(x) u_1(x)=\int_a^b K(x,y) u_2(y) dy$$ where $g(x)$ and $K(x,y)$ are known and $u_1(x)$ and $u_2(x)$ are complex ...
2
votes
0answers
20 views

Solving equation set with boolean operators and very specific format

I have to write a program to solve a set of equations like the following (+ is XOR and * is ...
2
votes
0answers
27 views

$\sum (\sqrt{x_k-k^2}-k)^2=0$ implies $x_k=2k^2$?

Let $x_1,x_2,\ldots,x_n$ be reals numbers such that $$\sum_{k=1}^n k\sqrt{x_k-k^2}=\frac12\sum_{k=1}^n x_k$$ Find all possible $n$-tuples of solution. So, I got the following solution from ...
2
votes
0answers
75 views

Impossible System of Equations

This is from a competition: DMM Olympiad, Ural State University P4 I don't understand what the question means exactly (the first part, i.e. "exclude $x$ or $y$ from..." part). Does it mean "write $x$ ...
2
votes
0answers
39 views

suggestion for lyapunov function

Consider differential equation \begin{align}x'&=-t(x+y)\\ y'&=-y+x-y(y^2-6).\end{align} Can some one suggest a lyapunov function for it. I have examined $V(x,y)=x^2+y^2$ , ...
2
votes
0answers
81 views

Solving systems of polynomials with an oracle

I need to solve a system of polynomials. Let the variables be $x_1, \dots, x_n$, and let the polynomials be $f_1, \dots, f_n$ Let's say we have these conditions we can already assume: there are ...
2
votes
0answers
36 views

Solving a system of two linear PDE: $u_x+v_x +u_y=0$ and $v_x+u_y-{1\over 2} v_y=0$

trying to solve the following cauchy problem: $$u_x+v_x +u_y=0\\v_x+u_y-{1\over 2} v_y=0\\u(x,0)=1-x,v(x,0)=x$$ my solution is: 1. multiply each equation by $t_1,t_2$ and sum the two equations like ...
2
votes
0answers
65 views

Study of a system of differential equations

I'm asked to study everything that is possible to know about the sytem$$\begin{cases}x'=x^2-y^2\\y'=2xy\\z'=-z\end{cases}$$ My questions here is, how much can be know about it?, how do I know I ...
2
votes
0answers
85 views

Computing a “cheap” upper bound on the norm of the solution to a linear system

Consider the linear system $A x = b$, where $A$ is an invertible, $n \times n$, real matrix. I would like to compute a "cheap" upper bound on the (p-)norm of the solution; i.e. $\|x\|_p$. One can, of ...
2
votes
0answers
105 views

Math software for plotting phase portraits

I'm looking for math software which is possible to plot phase portraits for ODE and systems of differential equations. Is there a software which can create not only simple 2D phase portrait plots but ...
2
votes
0answers
40 views

Nontrivial solutions for a system of equations

Consider $t:[0,1]^2\to R$ that is differentiable a.e. and satisfies conditions (i)-(ii): (i) $$ \int_0^1 \frac{\partial t}{\partial t_1}(x,y)f(y\mid x)\,dy=0, \quad \forall x\in[0,1] \\ \int_0^1 ...
2
votes
0answers
91 views

Combining two differential equations

I have two differential equations that are connected by an equation, $L_1\frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1=\frac{dV}{dt}$ $L_2\frac{d^2I_2}{dt^2} + \frac{1}{C_2}I_2=\frac{dV}{dt}$ $I_1+I_2=I$ ...
2
votes
0answers
20 views

Existenence of the solution for a PDE-ODE system.

I have the PDE-ODE system below: $\frac{\partial c}{\partial t}= D \Delta c - \eta \nabla.(c\nabla v)+g(c,v)$ $\frac{dv}{dt}=-\alpha cv+\xi(c,v)$ with initial conditions and Neumann boundary ...
2
votes
0answers
40 views

Find a reduced echelon basis from a reduced echelon matrix.

The reduced row matrix was this ---> $\begin{pmatrix}1&2&0&1&0\\0&0&1&3&0\\0&0&0&0&1\\0&0&0&0&0&\end{pmatrix} = 0$ So i computed ...
2
votes
0answers
82 views

Eliminate variable in trigonometry equations

Say you have the equations: \begin{align} -S_1\sin\left(2\psi+\theta\right)+S_2\cos\left(\psi\right)&=S_3\\ S_1\cos\left(2\psi+\theta\right)+S_2\sin\left(\psi\right)&=S_4 \end{align} or ...
2
votes
0answers
49 views

Solving systems of equations with trigonometric terms

I am trying to solve (or rather find the least squares solution for) a system of equations with trigonometric terms in them. The system consists of pairs of equations of the form $a_1 \cos\theta - ...
2
votes
0answers
59 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' = ...
2
votes
0answers
55 views

How do I solve this question without solving for the functions?

The problem goes as follows: $$\begin{aligned} \frac{d y_1}{dt} &= -ay_1 \\ \frac{d y_2}{dt} &= -by_2 -\frac{dy_1}{dt} \\ y_1(0)&=M \\y_2(0)&=0 \end{aligned}$$ where ...
2
votes
0answers
255 views

Gauss-Jordan is ALWAYS consistent with Cramer`s rule .

When using Gauss - Jordan elimination to solve a system of linear equations, is the solution you get after obtaining a matrix in Reduced Row Echelon Form THE solution or is there any chance that not ...
2
votes
0answers
144 views

Using variation of parameters, how can we assume that nether $y_1$, $y_2$ equal zero?

Here's the problem I have been given: Use the method of variation of parameters to find a general solution of the following differential equation: $$ a_1y''+a_2y'+a_3y=f(x) $$ And linearly ...
2
votes
0answers
58 views

Crossing Orbits

I have a question here that I am stumped with. The path of an orbit of a planet around a distant sun is $2K^2 + 2I^2 = 50$. The planet orbits the sun at roughly $900$ million kilometers. The path of ...
2
votes
0answers
53 views

Problem reduced to analyzing solutions of a family of nonlinear systems of equations

This was posted on mathoverflow about two weeks ago and I got no response so I'm asking here in case anyone has any ideas. Original post is here. I was able to reduce a research problem relating to ...
2
votes
0answers
52 views

Solving the algebraic equations .

I am working with an equation to find the singular points in $\mathbb P^2 (\mathbb C)$ . Basically after taking the partial derivatives and doing some manipulations it reduces to $$y^2 + (2-k)xz ...
1
vote
0answers
25 views

Solving System of Linear Equations

These are the two known equations: $$(I_2+I_3)-\frac{I_1+I_4}{I_1+I_2+I_3+I_4} = \frac{2x}{L}$$ $$(I_2+I_4)-\frac{I_1+I_3}{I_1+I_2+I_3+I_4} = \frac{2y}{L}$$ where I know the values of $(x,y,L)$. How ...
1
vote
0answers
64 views

Simple $\{-1,0,1\}$ equation set

I'm trying to find the shortest path, getting from $x=0$ to $x=k$ in a certain problem, where I can slowly accelerate and decelerate. It comes down to finding the smallest $n$ and set of values ...
1
vote
0answers
21 views

How to find value of an unknown in matrix to make system of linear equations consistent

I'm currently stuck on this question relating to finding the unknown in a matrix so that the system of linear equations is consistent. I need to solve for $\lambda$. My first instinct is to try and ...
1
vote
0answers
29 views

Expressing the Solution to a System of Differential Equations

My professor wrote the solution to a system as $$X = C_1 \begin{bmatrix}1 \\2 \end{bmatrix} e^{\lambda_1t} + C_2 \begin{bmatrix}3 \\4 \end{bmatrix} e^{\lambda_2t}$$ Where the column vectors are the ...
1
vote
0answers
30 views

Converting second order system into first order system (ODE)

The second order equation $\frac{d^2\vec{x}}{dt^2} = A\vec{x}\ + \vec{g}(t)$ models an earthquake's effect on a 7-story building. Let $x_j(t)$ be the displacement of the $j$th floor with respect to ...