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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5
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100 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
121 views

System of 3 equations

I am doing thermal calculation in electronics and when trying to device a general formula for equivalent system resistance to air flow of a part of real system, I ended with this system of three ...
4
votes
0answers
30 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
89 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
41 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
50 views

Analytic solution for system of Digamma function equations

Peace be upon you, There is a while, I am seeking for the analytic solution of $\alpha$ and $\beta$ in this system \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ ...
3
votes
0answers
48 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
67 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
38 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
34 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
29 views

Method of characteristics for systems of PDE (vs. Lewy's example)

Main question: How does the method of characteristics generalize for systems of first order PDE, as opposed to scalar PDE? Namely, is there such a generalization at all, and if so what information ...
2
votes
0answers
21 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
56 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 a system of equation and finding the largest possible value of one of the variables

This problem comes from question 5 in the PUMAC Algebra A competition (link here): Suppose $w, x, y, z$ satisfy $$w+x+y+z=25$$ $$wx+wy+wz+xy+xz+yz=2y+2x+193$$ The largest possible value of $w$ can ...
2
votes
0answers
30 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} = ...
2
votes
0answers
79 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 ...
2
votes
0answers
34 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
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' = ...
2
votes
0answers
48 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
74 views

Find basis of solutions of this linear system.

I am supposed to find basis of the subspace of vector space $ \mathbb{R}^{3} $ of solutions of this linear system of equations: $y = \left\{ \begin{array}{ll} x_{1}+2x_{2}-x_{3}=0 \\ ...
2
votes
0answers
106 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
55 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
49 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 ...
1
vote
0answers
17 views

Verification - Matrices and Linear Equations Part 1

Would just like to verify my Answers and bounce off my ideas and thinking with someone as I feel quite alone in this course. I am usually great at maths and enjoy it but these matrices and linear ...
1
vote
0answers
32 views

System of quadratic equations for a tetrahedron

I know the dimensions of the base of a tetrahedron and the angles between the non base sides at the apex. I want to know the lengths of the three non base sides. Let the base's corner points be $A, ...
1
vote
0answers
58 views

Long-time asymptotic behaviour of a system of two ODEs

We have the following nonlinear ODE: $$ f' = af-bg -(f+g)^k \bigl(f'(0) +g'(0)\bigr) + f'(0), $$ $$ \bigl(G-T(x)\bigr) g' = -af+bg - g'(0), $$ where $a,b,k,G$ are constants, $f'(0)$ and $g'(0)$ are ...
1
vote
0answers
44 views

intersection of a line and plane on a 3-sphere

Suppose I have two 4D points, $\mathbf{a}=(a_1,a_2,a_3,a_4)$ and $\mathbf{b}=(b_1,b_2,b_3,b_4)$, that both lie on a unit 3-sphere (i.e. unit distance from origin). In addition, I have a 2-D plane that ...
1
vote
0answers
22 views

Removing parametrization from a system of equations

Consider the following system : $$ \begin{aligned} \frac{d^2t}{d\lambda^2} &= -f\left(t\right)\frac{d t}{d \lambda}\frac{d t}{d \lambda} -A\frac{d g\left(t,x\right)}{d \lambda}\frac{d t}{d ...
1
vote
0answers
23 views

Solution of a DAE system of two ODE of second degree

I should solve the following DAE system: $$\ddot{x}(t)=-\alpha y(t)$$ $$\ddot{y}(t)=\beta x(t)$$ with the conditions: $x(t)\ge0$, $y(t)\ge0$ and $x(t)+y(t)=N$ with $N\gt 0$. I'm able to solve the ...
1
vote
0answers
22 views

A nonlinear system of equation

In the real numbder set: $x,y,z$ are variable, $a_i,b_i,c_i,d_i$ is given ($i\in\{1,2,3\}$) What is the conditions for the following equations have solutions? $$a_1xy+b_1x+c_1y=d_1$$ ...
1
vote
0answers
45 views

Time needed to algebraically solve system of $15$ nonlinear equations with parameters

How long can I expect it will take to algebraically solve a system of $15$ nonlinear equations (without any numbers, only parameters), if I feed it into a computing software? I'm asking for symbolic ...
1
vote
0answers
21 views

System of ODEs and DAE system

Let us consider the following system of ODEs: $$ y' = f(y,z),\quad z' = g(y,z),\quad y(0) = y_0,\;z(0)=z_0 $$ and the following one: $$ y' = f(y,z),\quad 0 = g(y,z), \quad y(0) = y_0. $$ $f$ and $g$ ...
1
vote
0answers
66 views

Solving a system of linear ODEs

Based on my previous post, I have been stuck on this for a few hours now. I want to solve for $x$ and $y$ from the equation $$\frac{dx}{dt} + \frac{dy}{dt}=a-(b+c+d)y-bx.$$ The original two equations ...
1
vote
0answers
19 views

How do I generate some sample solutions for an underdetermined system?

I have a system of 379 linear equations and 6325 unknowns. Does anyone know of a tool that can generate some (non-negative) solutions that satisfy this system? I know there are infinitely many, but it ...
1
vote
0answers
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 ...
1
vote
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 ...
1
vote
0answers
60 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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
0answers
53 views

Frobenius Method For System of Differential Equations

I have a system of ODEs. Can you explain how to solve a system of ODEs using the method Frobenius expansions ? There are 5 ODEs which are coupled and 5 variables. $\omega\hat\rho + i\alpha V_z ...
1
vote
0answers
44 views

Phase portrait of DS with skew symmetric matrix

How should I draw phase portrait of DS: $x'=Ax$, where $$A=\left( \begin{array}{ccc} 0 & 1 & 0 \\ -1 & 0 & -2 \\ 0 & 2 & 0 \\ \end{array} \right)?$$ Eigenvalues here are $0, ...
1
vote
0answers
50 views

find $x$, given $\{c_ix = k_i + y_i\}_{i=[1,n]} $

Given $$c_1x = k_1 + y_1 $$ $$c_2x = k_2 + y_2 $$ $$\vdots $$ $$c_nx = k_n + y_n $$ where the values of $\{c_1 \ldots c_n \}$ and $\{ k_1 \ldots k_n \}$ are known, and $x, \{y_1 \ldots y_n \}$ are ...
1
vote
0answers
54 views

Decoupling system of two partial differential equations

If I have the following systems of PDE $$ u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0, $$ where ...
1
vote
0answers
78 views

Geometry aspect of a extreme value problem

In a plain with orthogonal coordinate $XOY$, set point $A(a,a)$, and $P$ is a point in function $y=\frac{1}{x}$,where $x>0$. If the distance between $P$ and $A$ is $2\sqrt{2}$.Find all $a$ ...
1
vote
0answers
38 views

Using Gauss-Jordan for an infinite-solutions system

I'm starting to get the hang of this Gauss-Jordan stuff - well, I have never done a system with infinite solutions, so I decided to try this one. You can scroll to the bottom instead to see my doubts ...
1
vote
0answers
45 views

Lanchester's war model optimization.

Suppose the Lanchester's war model: $f'(t)=-0.5g(t)+x\sin^2(t)$ $g'(t)=-0.5f(t)+\cos^2(t)$ with $f(0)=g(0)=2$. How to estimate how small $x$ can be in order to make $f(t)$ won't reach $0$ on the ...
1
vote
0answers
16 views

A method of calculation coordinates in order to implement it to a code language!

lets say that we have three points A(xa,ya,za), B(xb,yb,zv), C(xc,yc,zc) with known coordinates in 3d space. Is there a method to calculate the coordinates (x,y,z) of another point D for which the ...
1
vote
0answers
85 views

Algebraic manipulation of Lyapunov function

I have a problem I would like some feedback on. I have spent 6 hours on it examining various techniques (numerically and analytically). I need to find the values of $k$ for which $x^2+ky^2$ is a ...
1
vote
0answers
31 views

Are there methods to solve a system of coupled integral equations?

I was wondering if there were methods to solve a system of coupled integral equations. The example case I am thinking about is $$f(x)=g(x)+\int_a^xf(x^\prime)h(x^\prime) dx^\prime$$ ...
1
vote
0answers
84 views

Solving a Large band system using Gauss-Seidel Iteration

Sorry for my english. I have to solve the following band system using Gauss-seidel iteration program in matlab. $$ \begin{array}{cccccc} 12x_1&-2x_2&+x_3&&&=&5\\ ...