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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1
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1answer
26 views

How can I write a set of equations in summation form?

I have a system of equations as follows: \begin{align} & A_1^{11} + A_1^{12} + A_1^{13} + \cdots + A_1^{1n}=X \\[8pt] & A_1^{21} + A_1^{22} + A_1^{23} + \cdots+ A_1^{2n}=X \\[8pt] & ...
0
votes
0answers
15 views

Counting the number of roots of multivariate polynomials?

The equation of a circle is well known $$(x-x_0)^2+(y-y_0)^2 - r^2 = 0$$ It has a solution all along the circle with midpoint $(x,y) = (x_0,y_0)$. We also know that $ab = 0$ whenever any of $a$ and/or ...
2
votes
1answer
451 views

Solving a System of Linear Equations (k value for infinite, unique and no solutions)

$$x(k+2) + y(k-1) + z(k) = 2$$ $$y(k+2) + 2z(k) = 0$$ $$ z(k^2 + k -2) = k + 2$$ Determine the values of k for which the system has: Exactly one ...
0
votes
1answer
24 views

solving a pair of simultaneous equations

I have a rather messy pair of simultaneous equations, which I need to solve for x: ...
1
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2answers
37 views

So many logs with different bases

$ \large { 6 }^{ \log _{ 5 }{ x } }\log _{ 3 }( { x }^{ 5 } ) -{ 5 }^{ \log _{ 6 }{ 6x } }\log _{ 3 }{ \frac { x }{ 3 } } ={ 6 }^{ \log _{ 5 }{ 5x } }-{ 5 }^{ \log _{ 6 }{ x } }$ The sum of ...
7
votes
2answers
16k views

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order ...
1
vote
2answers
40 views

Number of integral solutions for an equation

How do we approach this kind of problem of finding number of positive integral solutions to $$\frac{1}{x}+\frac{1}{y} = \frac{1}{n!}$$ Here $n$ is given.
2
votes
1answer
72 views

A seemingly-trivial divisibility conjecture

While working on another problem, I stumbled on the following divisibility claim. Conjecture: No integers $a,b,c,d$ satisfy all of the following conditions: $a^2+b^2-c^2-d^2 = 2(ad-bc)-1$; ...
0
votes
1answer
6 views

How is Dulac's Multiplier selected?

I'm aware that Dulac's (Negative) Criterion states that a system of differential equations of the form $x' = f(x,y), \; y' = g(x,y)$ has no periodic orbits in the plane if we can find some function ...
1
vote
1answer
35 views

Solve $A^kx=b$ system using $LU$

I have the system $A^kx=b$ and the $LU$ factorization $A=LU$. How can I solve the system without actually calculating $A^k$?
0
votes
0answers
20 views

Finding the General Solution for a System of Differential Equations with Complex Eigenvalues

I think I might just be having trouble with formatting my answer, because I'm fairly sure my work is right up until this point. The question asks to find the general solution to $$X'= ...
0
votes
0answers
15 views

Euler explicit and semi-implicit

I am given a simple dynamic system with an initial condition: $a(t) = 0.9 - 0.1v(t)$ $v(0) = x(0) = 0$ I want to calculate $x(1)$ with a time step of $\Delta t = 1$ using Euler explicit and semi ...
0
votes
0answers
17 views

Nonlinear equations algorithm - Newton method

Some time ago I posted a question regarding the simple case of finding the intersection point when I have only two functions, and with your help I found an answer. It was this case: $f(x) = a + ...
1
vote
1answer
35 views

Systems of equation

Find non-negative solutions of systems of equations: $$\begin{cases} x^2y^2+1=x^2+xy \\ y^2z^2+1=y^2+yz \\ z^2x^2+1=z^2+zx \end{cases} $$ My work so far: 1) $(1;1;1) - $ solution. 2) ...
1
vote
1answer
97 views

Solve this systems to condition $3x^3(x+1)^2=2y^2(z+3)^3$

if $x,y,z$be postive real numbers, solve systems of this following equation $$ 3x^3(x+1)^2=2y^2(z+3)^3\tag{1}$$ $$3y^3(y+2)^2=2z^2(x+1)^3\tag{2}$$ $$3z^3(z+3)^2=2x^2(y+2)^3\tag{3}$$ My approach is ...
-1
votes
1answer
40 views

Finding a and b from $a+b/3 = 1$ and $a/2+b/4=3/5$

I have two equations of which I need to solve for $a$ and $b$. $$ a+b/3=1\\ a/2+b/4=3/5 $$ Find $a$ and $b$.
0
votes
0answers
13 views

Sum of triangular matrices system

I was wondering if there is a nice way to solve the following linear system of equaitons: $(A+B) x = b$, where $A$ is an upper-right triangular matrix (all elements higher than the main diagonal are ...
1
vote
0answers
23 views

System of Nonlinear Equations (sum of powers)

I want to show the only solution to the following system of equations is the trivial one ($x_{i} = 0$). I don't know if this is true, but I think it should be. Let $x_{i} \in \mathbb{C}$ for $1 \le i ...
0
votes
1answer
44 views

exponential equation system without log [duplicate]

How should I solve this equation system without using logarythms,using just a simple method? (E.g. turning it into a quadratic one using t) $$\left(\frac{3}{2}\right)^{x-y} - ...
3
votes
4answers
118 views

Quick way to solve the system $\displaystyle \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} = \frac{65}{36}$, $xy-x+y=118$.

Consider the system $$\begin{aligned} \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} & = \frac{65}{36}, \\ xy -x +y & = 118. \end{aligned}$$ I have solved it by ...
1
vote
2answers
25 views

Solving system of non-linear equations.

So I'm trying to find the stationary points for $$f(x,y,z) = 4x^2 + y^2 +2z^2 -8xyz$$ Setting the partial derivatives to zero leads to: $$x-yz=0 \\ y-4xz=0\\z-2xy=0$$ Substiting $z=2xy$ into the ...
3
votes
2answers
47 views

The condition about some positive real numbers can be written as the sum the nearby two

Given $n$ positive real numbers $x_1,...,x_n$. What is the condition that they can be written as $$x_1=y_1+y_2$$ $$x_2=y_2+y_3$$ $$\ldots$$ $$x_n=y_n+y_1$$ where $y_1,\ldots,y_n$ are also some ...
0
votes
3answers
39 views

Real problems solved with systems

Can anybody tell me where can I find some REAL problems (i.e. form real life) that can be solved using a 3x3 system of linear equations? Or, can anybody give me an example? A solution could be a ...
1
vote
1answer
25 views

Nonlinear equations systems

Can anybody help me to find a system with 3 equations and 3 unknowns and a bounded domain D = [a,b]x[c,d]x[e,f] such that the system has an unique solution in D? Also, i need nice equations, because ...
0
votes
3answers
565 views

software to solve system of nonlinear equations

I am looking for a software to solve system of nonlinear equations. It would be great if the software can satisfy the following requirements It can support symbolic computation. It deals well with ...
0
votes
0answers
36 views

How to solve non-symmetric, generalized saddle-point problem

From a discretization of an incompressible Navier-Stokes equation I get the following matrix $$M = \begin{pmatrix}A & B_1^\intercal \\ B_2 & C\end{pmatrix}$$ Because my density is ...
-3
votes
0answers
34 views

Advance Algebra [on hold]

I have an interesting problem if anyone can solve it: Harry rides by train to go home and arrives at the railway station every day at 18h00. His wife travels by car to pick him up at the station and ...
0
votes
1answer
65 views

System of equation $x+y+z=2007; xyz=14000$

I have to solve the system of equations $$\begin{cases} x+y+z=2008,\\ xyz=14000, \end{cases}$$ where $x,y,z$ are positive integers such that $1\le x \le y \le z \le 2000.$ My work so far: ...
0
votes
1answer
24 views

Consistency of system of linear equations

Find when the equations $$\begin{cases}x + y - 2z = 0\\ax + by + cz = 0\\bx + cy + az = d\end{cases}$$ are consistent and solve them completely when they are consistent. I have tried the ...
0
votes
0answers
34 views

Solution of seemingly simple system of equation

I wand to find the solutions $p_H, p_L$ implied by the following two equations: (I) $\frac{(1-\lambda)(p_L-c_L)}{p_H \frac{q_L}{q_H}-c_L} = \frac{\lambda(p_L-c_L)}{p_H-(q_H-q_L)-c_L} - \lambda$ (II) ...
2
votes
6answers
119 views

Solve the system of equations: $a+b+c=2$, $a^2+b^2+c^2=6$, $a^3+b^3+c^3=8$ [on hold]

If we have \begin{cases} a+b+c=2 \\ a^2+b^2+c^2=6 \\ a^3+b^3+c^3=8\end{cases} then what is the value of $a,b,c$?
0
votes
0answers
17 views

Solve a system of simple (but not linear) equations

There are 24 variables and 24 equations in the system: $$ i=0,1,2,3.\\ variables: s_i, t_i,a_m \; (m=0,...,15)\\ constants: b_{ni} \; (n=0,...,5)\\ a_0t_i+a_1s_i=b_{0i} \\ a_2t_i+a_3s_i=b_{1i} \\ ...
1
vote
2answers
52 views

The sum of two numbers is 5/9…

The sum of two numbers is $\frac{5}{9}$. The quotient of the two numbers is $1$. What is the product of $40$% of each number? The answer I got was $\frac{1}{81}$. I don't understand this - would ...
0
votes
1answer
18 views

how do you solve the simultaneous equations 2x + y = 9 and x - 2y = -8 using the matrix method?

i first convert the bases into a 2x2 matrix and then i multiplied the inverse matrix by the 2x1 e/f matrix of 9 (on top) and -8 (on the bottom) which gave me x = 5.4 and y = 5 however the answer ...
0
votes
1answer
22 views

Find the length of the longest diagonal of the bo

The total length of all $12$ sides of a rectangular box is $60$. The surface area of the box is given to be $56$. Find $(i)$ the length of the longest diagonal of the box $(ii)$ the volume of the box ...
0
votes
0answers
21 views

How to numerically minimize system of equations composed of data and smoothness terms, ensuring minimum solution norm

I need to find $g$ that minimizes: $$\sum_{v=0}^n (f+g_{v_{left}}-g_{v_{right}})^2 + \frac{1}{\lambda}\sum_{v=0}^m (g_{v_i}-g_{v_j})^2$$ where $f$ is constant and the sums are over pair of $v$ ...
3
votes
1answer
42 views

Symmetric system of equations problem

Solve the following simultaneous eqations on the set of real numbers: $$a^2+b^3=a+1$$ $$b^2+a^3=b+1$$ I have found two trivial solutions: $$a=b=1$$ $$a=b=-1$$ but I can't prove that there are no ...
1
vote
0answers
21 views

Reducing a system to first order

Convert the following to a first order system $$x''(t) = k_x(x(t) - y(t))^{-2}, \ \ y''(t) = k_y(x(t) - y(t))^{-2},$$ $$x'(0)=v_x, \ \ y'(0) = v_y, \ \ x(0) = x_0, \ \ y(0) = y_0.$$ I know how to ...
-2
votes
1answer
32 views

Linear Algebra - Row echelon form [closed]

Find two different row echelon forms of: $$\left(\begin{matrix} 1&4\\ 3&11 \end{matrix}\right)$$ this exercise shows that a matrix can have multiple row echelon forms. I konw it is easy, ...
0
votes
0answers
8 views

Closed-form solution for system of equations for finding a critical point

I am trying to find a critical point of a function $\mathbb{R}^d \to \mathbb{R}$ by setting its gradient to zero. I would like to solve the follwoing system of equations. $$\frac{1}{1 - \sum_{j=1}^d ...
4
votes
2answers
128 views

Prove that this system of linear equations generates $\left| \left( \begin{matrix} 1/2 \\ n \end{matrix} \right) \right|$ as a solution?

This infinite system of linear equations: $$ \begin{array}( 2x_1=1 \\ 3x_1+4x_2=2 \\ 4x_1+5x_2+6x_3=3 \\ \cdots \end{array} $$ In other words, this is particular case of a system: $$ \begin{array}( ...
1
vote
3answers
63 views

What does x equivalent to 2 mod 15 mean?

I came across the following question: Consider the following system of equivalences of integers. $$ x \equiv 2 \bmod{15} $$ $$ x \equiv 4 \bmod{21} $$ The number of solutions in $x$, where $1\le ...
4
votes
2answers
64 views

Why is a polynomial $f(x)$ sum of squares if $f(x)>0 $ for all real values of $x$?

If a polynomial $f(x)>0 $ for all real values of $x$, then $f(x)$ is sum of squares. Why is this true ? I understand that the roots of this $f(x)$ will be complex and hence will exist as ...
2
votes
1answer
682 views

Numerical techniques for solving systems of first-order semi-linear hyperbolic PDEs in two variables

I need to solve such systems of PDEs numerically and I'm wondering what the "standard" method (if there is one) is. The systems I'm interested in have the form: $\frac{\partial F_i}{\partial t} + ...
1
vote
1answer
575 views

solving a non-linear (trigonometric) system of equations with two equations and two variables

I'm trying to solve the following system of equations: $$l_1*sin(\alpha)=l_2*cos(\gamma)+l_3*sin(\beta)$$ $$l_2*sin(\gamma)+l_1*cos(\alpha)=l_3*cos(\beta)+l_4$$ with the unknowns $\beta$, $\gamma$ ...
-1
votes
1answer
27 views

Rewriting system as a set of first order equations.

What I'm given: $$x'' = x' + y' + x + y$$ $$y'' = 2x' + 3y' + 3x + y$$ $$z=x'$$ $$w=y'$$ My solution: We know that $z'=x''$ and $w'=y''$. We can write: $$z'=z+w+x+y$$ $$w'=2z+3w+3x+y$$ I'm not ...
17
votes
2answers
495 views

Does this system of simultaneous Pell-like equations have any non-trivial positive integer solutions?

Let $a,b,c$ be positive integers satisfying \begin{align} 2a^2-1 &= b^2, \\ 2a^2+1 &= 3c^2. \end{align} The trivial solution is $(a,b,c)=(1,1,1)$. Are there others?
1
vote
0answers
36 views

Resolve integral equations

There is a way to solve this problem? Let be $[a,b]$ an interval where $a$ is finite but $b$ can also be infinity. Find a function or a distribution $h(u,s)$ for $u,s \in \mathbb{R}$ such that for ...
0
votes
1answer
24 views

How do I solve this 3-D system of linear equations using Gaussian elimination?

I have the following system of equations: $x+2y+3z = -6$ $2x - 3y - 4z = 15$ $3x + 4y + 5z = -8$ I came up with this: $x + 2y + 3z = -6$ $-7y - 10z = 18$ $5x + y + z = 7$ Can you tell me the ...
0
votes
0answers
9 views

When does a system of n symmetric polynomials in n variables have exactly one solution over C up to permutation?

I was slightly amused that if I never learned about polynomials and was asked if Vieta's system of equations has exactly one solution up to permutation, the solution would be to develop polynomials in ...