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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2
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2answers
40 views

Conditional Extremum, need help finding the extreme points in calculation.

Find the conditonal extremums of the following $$u=xyz$$ if $$(1) x^2+y^2+z^2=1,x+y+z=0.$$ First i made the Lagrange function $\phi= xyz+ \lambda(x^2+y^2+z^2-1) + \mu (x+y+z) $, now making the ...
-1
votes
2answers
20 views

To test following system of linear equation for equivalency

Let F be field of complex numbers I have two system of equations $x_1 - x_2 =0 $ $2x_1 + x_2 =0$ And $3x_1 + x_2 -0$ $x_1 + x_2 =0$ The definition says that each if equation in first system ...
0
votes
1answer
21 views

The phrase “periodic boundary conditions” for a two-variable PDE

I'm currently working on trying to solve a system of PDE's of the form $c_t=D_x(c_{xx} + c_{yy})+K_1 c + K_2 d$ $d_t= D_y(d_{xx}+d_{yy})+K_3 c + K_4 d$ that has "periodic boundary conditions" on a ...
0
votes
0answers
28 views

Numeric solution for a non-linear system

It has been a while I have not practiced mathematics but I should have enough background to get your answers if well detailed. I have an n-by-n matrix, let's call it D, where dij represents the ...
0
votes
1answer
32 views

Advice to solve a system of 8th order univariate polynomials

I am struggling to solve a least square problem in which the tedious part is the initialization. Grid search methods are out of question. The initial problem I've stated my problem in a previous ...
0
votes
1answer
31 views

Find the solution to the system (not linear)

Find all $(x, y, z) \in \mathbb{R^3}$ satisfying: $$x^2 + 4y^2 = 4xz \tag1$$ $$y^2 + 4z^2 = 4xy \tag2$$ $$z^2 + 4x^2 = 4yz \tag3$$ This is a very difficult problem. I added $-4(1) + (3)$ to ...
2
votes
1answer
19 views

system of equations with $n$ equations and $2^k n$ unknowns

I have a system of equations with infinitely many solutions. I would like to find a "nice" way to write down an explicit solution. Here, $n,k\geq 1$ are integers, we have $x_1,x_2,\dots, x_{2^k n}$ ...
0
votes
2answers
29 views

Explain how solution got $c_1$ and $c_2$

Can someone explain how the solution manual got $c_1$ and $c_2$ in this:
0
votes
0answers
20 views

Solving a system of differential-algebraic equations

I am seeking for references around the topic of solving this type of differential equation system : $$ \left\{ \begin{array}{ll} \partial_x y_{i+1}(x) = y_{i+1}(x)-y_{i}(x) \\ y_{i+1}(x) = y_i(f(x)) ...
1
vote
0answers
31 views

Convert a 2D autonomous ODE system into a 1D system?

Suppose I have two equations: $$\frac{dx}{dt} = x(2-x-y),\, \frac{dy}{dt} = ky(2-ax-by)$$ that together form a 2D ODE system ($x$ plotted against $y). K, a$ and $b$ are all independent positive ...
0
votes
0answers
31 views

Conormal Points Parabola

Let the line $lx+my=1$ cut the parabola at $y^2=4ax$ in the points A and B.Normals at A and B meet at a point C. Normal from C other than these two meet at D.Then coordinates of D are? I tried to ...
0
votes
0answers
29 views

Systems of equations of the form $\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a$

Is there any theory that deals (directly or not) with systems of equations of the form $$\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a,$$ where $a \in \mathbb{R}^3$ is known, $v_i, v_j \in ...
0
votes
1answer
37 views

Optimizing for the minimum relative distance in a given situation?

I have primarily been working on this problem for quite some time now; the level of the problem is introductory calculus w/ optimization problems. The situation is as follows: Ship A sails due ...
0
votes
0answers
29 views

Ordinary differential equation system with maximum value of x+y

I've got the following simple system of ODEs: x': x*(1-x-y) y': y*(1-a* y-b* x). Plotting the system in a x' y' plot is not fully satisfying to me as what I want to achieve is that the sum of x and ...
5
votes
2answers
92 views

How to prove whether the equation set has a unique solution?

\begin{eqnarray} \begin{cases} \sin A \sin C-(\sin B)^2=0 \cr AC-B^2=0 \cr A+B+C-\pi=0 \cr A>0,B>0,C>0 \end{cases} \end{eqnarray} How to prove whether the equation set has a unique solution ...
1
vote
3answers
26 views

multiplying term on sum

Say I know the following relation holds $$ \sum_i f_i + \sum_i g_i = 0 $$ Now I multipy both sides with a set of vectors $\mathbf v_i$. Will it still be true that $$ \sum_i f_i \mathbf v_i + \sum_i ...
1
vote
2answers
55 views

Solving Systems of Linear Differential Equations by Elimination

For a homework problem, we are provided: $\frac{dx}{dt}=-y + t$ $\frac{dy}{dt}=x-t$ Putting these into differential operator notation and separating the dependent variables from the independent: ...
1
vote
0answers
25 views

I need help with creating linear equations from multiple points on a graph

How do you create a linear equation from multiple points on a graph I am working on a question where the points are $(-3,8),(2,5)$ and $(7,2)$ and I need to find out how to create a linear inequality ...
1
vote
1answer
27 views

solution of a system of equation

Let $A\in M_{m\times n}(\Bbb R)$ and let $b_0\in \mathbb R^m$. Suppose that the system of equations $Ax=b_0$ has a unique solution. Which of the following is true? $Ax=b$ has a solution for every $b ...
1
vote
2answers
123 views

Solve a system of two nonlinear equations

$$ \begin{cases} x^2 - y^2 + 12y - 21 = 0\\ 2x^2 + y^2 + 2xy + x = 0 \end{cases} $$ I've tried the change of variables: $u = x + y$, $v = x - y$ After it I've got: $$ \begin{cases} uv + 12\frac{u - ...
2
votes
1answer
46 views

A system of linear equations with 3 variables such that its solution set is: $\{(a,b,c)|a^2=b \}$?

Is there a system of linear equations with 3 variables such that its solution set is: $\{(a,b,c)|a^2=b \}$? It's enough to show that for one equation: $Ax+By+Cz=D$ the solution set doesn't work. ...
1
vote
2answers
35 views

Solve the following in vector form:

So i did a substitution to solve the system normally, and got $x=17.67$ $y=9.67$ $z=10.67$ Where I am stuck is how to represent something like this in a vector form, maybe my solution was wrong ...
1
vote
1answer
39 views

Rearranging An Equation To Solve?Can't?

How would I rearrange the equation: $$a=b^{(c/d)}$$ to find c?
1
vote
0answers
39 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 ...
1
vote
0answers
25 views

canonical form of parabolic-type PDE involving exp(x) and ln(x)

The attached picture skips a lot of the work, but I've worked this problem at least 6 times in the last 8 hours, still getting stuck at reducing to canonical form - that is, trying to solve for x and ...
0
votes
2answers
32 views

3 Variable System of Equations When All Set to Zero

So I'm doing an a bit of a pre-assessment for something, and I feel like I am missing something on this question: Now I know how to solve a normal 3 variable system, but with this they are all set ...
1
vote
1answer
52 views

Analytic solution of a system of four second order polynomials

Can I systematically solve in $\mathbb{R}^4$ the following system without using Grobner basis algorithm ? If not, can I find the exact number of solutions ? $$ \begin{equation*} \left\{ ...
0
votes
1answer
26 views

Solution for associated homogeneous linear system

If an inhomogeneous system of linear equations has an associated homogeneous system that has only the trivial solution, then how can I show that the inhomogeneous system has exactly one solution?
6
votes
1answer
163 views

Amount of solutions to the Diophantine equation of Frobenius

The Diophantine equation of Frobenius is any equation of the form: $$\sum_{i=1}^k a_i x_i = n$$ where the $a_i$'s are given and so are $k$ and $n$. I'm looking for an algorithm to compute the number ...
0
votes
1answer
62 views

Is it possible to create any combination of areas?

Given a point $P(x,y)$ in the unit square, two polygons Blue and Green are defined by drawing a 45-degree line through $P$ and creating polygons with the top-left and bottom-right corners, ...
1
vote
1answer
29 views

Behaviour of roots of a polynomial with function coefficients

Let $(-1+c_4(h))x^4 +c_3(h)x^3+c_2(h)x^2+c_1(h)x+c_0(h)=0$ be an equation with variable coefficients, depending smoothly on $h$. Also let $0\le c_4(h)\le 1-\epsilon$ for some $\epsilon>0$ and ...
1
vote
3answers
37 views

gaussian elimination to solve a question (using a paramter)

I want to solve : x2+x3=0 -x1 -x3=0 x1-x2 =0 I got the $x_1 = -t, x_2=-t, x_3=t$. But the book has $x_1 = t, x2=t, x3=-t$. ...
0
votes
2answers
29 views

Find system of equations such that

Find system of equations that will describe: a) plane $M \subset \mathbb{R}^3$ passing through the points $(6,1,-3), (1,5,1), (1,8,2)$ b) line $L \subset \mathbb{R}^3$ passing through $(1,2,-1), ...
1
vote
2answers
97 views

Solving a non-linear, multivariable system of equations

I'm researching the mathematics behind GPS, and at the moment I'm trying to get my head around how to solve the following system of equations: $\sqrt{(x-x_1)^2+(y-y_1)^2+(z-z_1)^2}=r_1$ ...
0
votes
1answer
52 views

Solving 4 unknowns with 4 equations all equal to zero

I have the following equations: $$\begin{align} a&=0.35a+0.35d\\ b&=0.65a+0.65d\\ c&=0.35b+0.35c\\ d&=0.65b+0.65c \end{align}$$ I know $b=d$, but where do I go from here? I have a ...
0
votes
2answers
53 views

Solve 5 unknowns with 5 equations

I have the following set of equations: $A = x_1x_2x_3x_4x_5$ $B = x_1x_3x_5 + x_1x_4x_5 + x_2x_4x_5 + x_2x_3x_4$ $C = x_3 + x_4 + x_5$ $D = x_1x_2x_3x_4$ $E = x_1x_3 + x_1x_4+x_2x_4$ Where A, B, ...
0
votes
1answer
63 views

How to solve the system of 4 equations of four unknowns

Solve this system of the four equations of four unknowns $a, b, c, d>0 $ $$ 165(a+b+c)=abc\tag1 $$ $$220(a+b+d)=abd \tag2 $$ $$297(a+c+d)=acd\tag3 $$ $$540(b+c+d)=bcd \tag4 $$ I tried to ...
3
votes
2answers
36 views

How to solve the given system of equations for $I_1, I_2, I_3$?

I have this system of equations: \begin{cases} I_1 = I_2 + I_3 \\ \epsilon_1 - I_1(R_1 + R_2) - I_2 R_3 = 0 \\ \epsilon_1 - I_1(R_1 + R_2) - I_3(R_4 + R_5) + \epsilon_2 = 0 \end{cases} I want to solve ...
0
votes
0answers
40 views

A simple non-linear equation.

Let ${\rm a}$ and ${\rm b}$ be two given vectors in ${\mathbb R}^n$. Find ${\rm u}\in {\mathbb R}^n$ and $x\in[0,\infty)$ such that $$ {\rm a} +{\rm b}x+\frac{1}{2}x^2{\rm u}=0, \quad \|{\rm u}\|=1 ...
0
votes
0answers
32 views

How can I find multiple solutions for a system of equations?

I'm writing a program for CheckIO.org that is supposed to return an array, $$ \begin{bmatrix} x\\ y\\ z \end {bmatrix} $$ , that satisfies the System of Equations $$ A \begin{bmatrix} x\\ y\\ z \end ...
2
votes
0answers
20 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 ...
1
vote
1answer
36 views

Find integral of the 2 by 2 system of ODE

We want to find a function $F(x(t),y(t))=c$ where $x(t),y(t)$ are solutions to the system $\begin{bmatrix} \dot x=\frac{t-y}{y-x} \\\dot y=\frac{x-t}{y-x}\end{bmatrix}$. Such a function $F(x(t),y(t))$ ...
2
votes
1answer
24 views

Add together simple equations

I have three equations $$ x = 20 \\ -x+a = 10 \\ y = 2 $$ Can I add these equations and get $$ x-x+a+y = 20+10+2 \\ a+y = 32? $$ If yes, what is the name of the rule applied?
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 ...
0
votes
0answers
14 views

Show the following system is not possible

Assume throughout that the base field is the prime field $\mathbb{F}_2$. I have two $n \times n$ matrices: $I_n$, the $n \times n$ identity matrix, and $C_n$ the matrix obtained from $I_n$ by shifting ...
0
votes
1answer
29 views

System of homogeneous linear equations issue

Solve the following system: $4x-12y+z=0\\ x-5y-z=0\\ -4x+12y+z=0$ So in matrix form it is $ \left(\begin {matrix} 4 & -12 & 1 \\ 1 & -5 & -1 \\ -4 ...
1
vote
1answer
23 views

Find base vectors and dim

Find base vectors and dim of a space described by the following system of equation: $$2x_1-x_2+x_3-x_4=0 \\ x_1+2x_2+x_3+2x_4=0 \\ 3x_1+x_2+2x_3+x_4=0$$ I did rref of the matrix and as a result i get: ...
1
vote
1answer
45 views

How can I solve the following exercise

How can I solve the following exercise $$φ_1(x)=e^x-\int_{0}^{x}φ_1(t)dt+4\int_{0}^{x}e^{x-t}φ_2(t)dt$$ $$φ_2(x)=1-\int_{0}^{x}e^{-x+t}φ_1(t)dt+\int_{0}^{x}φ_2(t)dt$$
0
votes
0answers
32 views

What are “symmetry arguments” in the context of solving systems of equations?

What and how are the "symmetry arguments" used to solve a system of equations? My text makes extensive use of this argument but do not provide and explanation of how it works or the definition of ...
0
votes
1answer
26 views

Solve $|x - z_1| = d_1 + y$ and $|x - z_2| = d_2 + y$ simultaneously for $x$ and $y$

Given the two equations $|x - z_1| = d_1 + y$ and $|x - z_2| = d_2 + y$ , and suppose that $z_1, z_2 \in \mathbb{R}$, $z_1 \neq z_2$ and $d_1, d_2, \in \mathbb{R}_{> 0}$ are all known reals, solve ...