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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6
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155 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 ...
5
votes
0answers
53 views

Why does the taylor expansion of a nonlinear system of differential equations exist if it has continuous second order partial derivatives?

My textbook states that for a nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y)$$ The system is locally linear in the neighborhood of a critical point $(x_0,y_0)$ whenever ...
4
votes
0answers
49 views

Why is $\frac d{dt}((\xi \alpha)^{-1})=\frac{-1}{(\xi \alpha)^2} \frac d{dt}(\xi \alpha) = \frac{\partial \lambda_2}{\partial w_1} \xi^{-1}$?

My question concerns the proof of Theorem 2 in §11.3 of PDE Evans: THEOREM 2 (Riemann invariants and blow-up). Assume $\mathbf{g}$ is smooth, with compact support. Suppose also the genuine ...
4
votes
0answers
57 views

Solve the system of equations with $x=y$

Solve the system of equations: $\left\{\begin{array}{l}\sqrt{x^2+(y-2)(x-y)}+\sqrt{xy}=2y\\\sqrt{xy+x+5}-\dfrac{6x-5}{4}=\dfrac{1}{4}\left(\sqrt{2y+1}-2\right)^2\end{array}\right.$ I used ...
4
votes
0answers
163 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
65 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
99 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
50 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
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39 views

An equation over a finite field

Suppose $x,y,z,w \in \mathbb{F}_{q^2}$, where $q=p^k$ for some prime $p$. Consider the system of equations $$ \left\{ \begin{array}{l} xy + zw = 0; \\ xy^q + yx^q + zw^q + wz^q = 0. \end{array} ...
3
votes
0answers
38 views

System of equations for operations

Given a system with multiple equations, where we know the values and the result, but not the operations between the values: \begin{cases} 3 ⊕ 5 ⊙ 2 = 13 \\ 7 ⊕ 2 ⊙ 4 = 10 \\ 4 ⊕ 3 ⊙ 3 = 9 \end{cases} ...
3
votes
0answers
67 views

Solving simultaneous equations in complex numbers

Given $z_1,z_2$ are complex numbers such that sum of their squares is a real number and $$z_1(z_1^2-3z_2^2)=2$$ and $$z_2(3z_1^2-z_2^2)=11.$$ I need to find the value of sum of squares of two complex ...
3
votes
0answers
105 views

System of symmetric equations

I was working on writing some problems for a contest, and I wrote the following system of equations: \begin{align*} x^2+yz&=259,\\ y^2+zx&=217,\\ z^2+xy&=203. \end{align*} Of course, ...
3
votes
0answers
135 views

Properties of polynomials that are polynomial conditions on the coefficients

There are many occasions where we can check whether a (set of) polynomial(s) $f_i$ satisfies certain properties, simply by evaluating a fixed polynomial on the coefficients of the $f_i$. Many times, ...
3
votes
0answers
66 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
37 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
48 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
105 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
43 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
64 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
267 views

Combining two differential equations

I have two differential equations that are connected by an equation, $L_1\frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1=\frac{dV}{dt}$ $L_2\frac{d^2I_2}{dt^2} + \frac{1}{C_2}I_2=\frac{dV}{dt}$ $I_1+I_2=I$ ...
3
votes
0answers
235 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
46 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
53 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
91 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
52 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
40 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
22 views

Solving A System Of ODE's On MAPLE 17

I Have the velocity fields for two vortices that are located at two different points ${\bf{x_1}}(x_1,y_1)$, ${\bf{x_2}}(x_2,y_2)$ $\vec{V_1} = (u_1,v_1) = \frac{\Gamma_1}{2\pi}\frac{1}{(x-x_1)^2 + ...
2
votes
0answers
39 views

Companion matrix of bivariate polynomial

A polynomial in one variable can be expressed as a companion matrix, of which the eigenvalues are the roots of the polynomial and which can be found by using e.g. QR decomposition or power iteration. ...
2
votes
0answers
34 views

Why can't we model periodic phenomena using a single autonomous differential equation?

I have the system below. It is used to model the interaction between predator and prey. $$x' = x-xy, y' = -y + xy$$ The solution curves are closed contours about the point $(1,1)$. I determined ...
2
votes
0answers
52 views

Are two linear system equivalent?

Let $A$ and $M$ be square matrices of size $s$ and $n$ respectively, let $k_i \in\mathbb{R^n}$ be column vectors for all $i=1,\ldots,s$. Denote $K=\left[ \begin{matrix} {{k}_{1}} \\ \vdots ...
2
votes
0answers
61 views

Finding Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem (SOLVED)

I need to find the eigenvalues and corresponding eigenfunctions of the following differential equation: $$y'' = -sy, \;\;\;\;y(0)=2y'(0), \; y'(1)=0.$$ I've already found the eigenvalues of the ...
2
votes
0answers
56 views

Solve general solution of system of first order partial differential equations

$$u_x + v_y = 0$$ $$v_x - v_y = 0$$ $$v_y - u_x = 0$$ We are instructed to solve the following system of first order equations. I have no idea where to begin. I have tried putting this into a ...
2
votes
0answers
235 views

Solving a system of exponential equations

$$2^{ x+y }=16\\ 3^{ x }-3^{ y }=24$$ Steps I took: $$\ln(2^{ x+y })=\ln(2^{ 4 })$$ $$(x+y)\ln(2)=4\ln(2)\Rightarrow x+y=4$$ $$\ln(3^{ x })-\ln(3^{ y })=\ln(24)\Rightarrow x-y=\frac { ...
2
votes
0answers
85 views

Finding The Summation of N term

I have a Two sequence A and B which are in a serious relationship which is as: where $x = \sqrt{2}$ and $y = \sqrt{3}$ I have to find : $a_k+b_k$ in general I came with this formula($a_n$ and ...
2
votes
0answers
42 views

Inverting an isometric projection?

I'm trying to invert a function that takes points on a 2-d plane to an isometric projection of that plane. This function is encoded as follows (as part of the Isomer library): ...
2
votes
0answers
66 views

Solving $-1=e^a-2e^{av}$ as part of a equation system

Problem Given $f_2(x)=e^{ax-b}+c$ with $x \in \left(0,1\right)$, I am trying to calculate the parameters $a,b,c$ in respect to the following constraints: $$ \begin{align} f_2(0) &= 0 \\ ...
2
votes
0answers
25 views

Locating a point in 2D using only differences between distances

Let $R_1,R_2,R_3,R_4,T$ be points on a 2D plane, as in this figure. $R_1,R_2,R_3,R_4$ are reference points with known positions. The goal is to find the position of $T$. $d$ is the Euclidean ...
2
votes
0answers
49 views

Is this system of inequalities (and equality) tractable?

I have some real parameters here. The $\mu_i$ - for $i=1,2,3,4,5$ - are 'convex coefficents' in that $\mu_i\geq 0$ and $\sum_{i}\mu_i=1$. The $x$ and $z$ are such that $x^2+z^2\leq 1$. The ...
2
votes
0answers
23 views

$L$-existential and $L$-diophantine

Could you explain to me the last sentence: "Whenever we want to stress dependence on the language, we will use the self-explanatory terms and $L$-existential and $L$-diophantine" ? What does ...
2
votes
0answers
24 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
83 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
34 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
151 views

Nonnegative solution of a linear system

Given three collections of parameters $\epsilon_1 > ... > \epsilon_N$, $(a_1,...,a_{N-1})$ and $(b_1,...,b_N)$ that satisfy the following conditions $\forall i, a_i \geq 0, ...
2
votes
0answers
55 views

least square solution of overdetermined system with additional unknown

I was hoping somebody could tell me the best way to solve the following overdetermined system for the scalars $x_{1}$,$x_{2}$ and $x_{3}$, where the C $3 \times 1$ vectors are unknown, $A_{i}$ is a $3 ...
2
votes
0answers
30 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
37 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
32 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
138 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
50 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
98 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 ...