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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4
votes
1answer
115 views

Does $A$ commute with $e^{\int A \: dt}$

I have been studying the linear system of the form: $$D_tX = AX + \textbf{b}$$ Where $A$ is not necessarily constant Suppose we aim to find an integrating factor $M$ such that: $$M[D_tX - AX] = ...
0
votes
0answers
25 views

Can I divide entire equations by each other when trying to solve a system of non-linear equations?

I have the following equations: $\frac{F}{(W-\phi)t}=\sigma_{_L}$ $\frac{F}{(W^2-\phi^2)t}=\tau_{_L}$ $\frac{F}{\phi t}=\sigma_{_L}$ $\frac{F}{\phi^2\pi/4}=\tau_{_P}$ I then solve for F ...
2
votes
2answers
35 views

A motorboat going downstream overcame a raft at a point A (Kinematics question)

A motorboat going downstream overcame a raft at a point A. $T$ = $60$ min later it turned back and after some time passed the raft at a distance $l$ = $6$ km from the point $A$. Find the flow ...
0
votes
1answer
34 views

Finding the equilibrium points of a 3D system

I've only done this for 2D systems, never for 3D. Find the equilibrium points for this system of equations (your answer will depend on the value of $N=S+I+R$, which you may assume to be a constant). ...
2
votes
1answer
42 views

if $A,B,C$ are real numbers such that ,${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $ and $A+B+C = 0 $ find the maximum value of $(ABC )^2$ [duplicate]

$$A,B,C$$ are real numbers such that ,$${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1 $$ and $$A+B+C = 0 $$ find the maximum value of ${ (ABC) }^{ 2 }$ I don't know how can I start to solve this ...
-1
votes
1answer
20 views

Difference of DAE and ODE [closed]

What are the differences between ODE and DAE? Is every ODE a DAE? Please explain with examples
0
votes
2answers
29 views

Reduce the number of solutions of a trigonometric equation system

Equation system Given the following system of trigonometric equations $ d_x=\sin{\alpha}\cdot\sin{\beta}\\ d_y=-\sin{\alpha}\cdot\cos{\beta}\\ d_z=\cos{\alpha}$ The unknowns are $\alpha, \beta$. ...
0
votes
1answer
8 views

Simplifying a complicated system of equations involving $\min$.

$o_n = 2n + 1$ $e_n = (2n)^2$ $r_i = \mathop{\min} \{ n \mid e_n \ge i\}$ $a_n = n + e_n$ $q_i = \left\lfloor \frac{i - a_{r_i}}{o_{r_i}} \right\rfloor \mod 4$ Is there another way to write $q_i$ ...
-1
votes
1answer
43 views

system of linear equations in a basic model [closed]

Consider an farm where only wheat and cow (meat) is produced. For the sake of simplicity we will measure the amount of wheat and cow (meat) produced in tons (T). In order to produce w T of wheat one ...
1
vote
3answers
58 views

How to solve a system of two equations in three unknowns

$x+y=5$ $2x+y-3z=12$ I know that in order to solve three unknowns three equations are needed, so I'm unsure if this can be solved or if different techniques (apart from the usual ...
2
votes
1answer
20 views

CRT - non-linear system of equations

I don't know how to solve system of equations using CRT when there is some quadratic/cubic variable. For example: System 1: $$\boxed{x^3 \equiv 1 \pmod{3}}$$ $$12x \equiv 9 \pmod{15}$$ ...
1
vote
3answers
61 views

How do I solve the system of congruences $x \equiv 1 \pmod{3}$, $x \equiv 2 \pmod{4}$, $x \equiv 4 \pmod{5}$, $x \equiv 2 \pmod{7}$?

Sorry for the weird title, but I don't know how else to phrase this. Say I have a set of numbers, and the remainders each get by dividing them by a certain number. For instance: $x \equiv 1 ...
1
vote
1answer
43 views

Finding the Coefficient Matrix of a Spring-Mass System

So as part of a class in numerical linear algebra, we're exploring the topic of banded matrix system. I've come across a problem that involves Hooke's Law, but I'm having a little difficulty ...
2
votes
3answers
37 views

Find all possible values of $c^2$ in a system of equations.

Numbers $x,y,z,c\in \Bbb R$ satisfy the following system of equations: $$x(y+z)=20$$ $$y(z+x)=13$$ $$z(x+y)=c^2$$ Find all possible values of $c^2$. To try to solve this, I expanded the equations: ...
7
votes
2answers
59 views

Solve system of simulataneous equations in $3$ variables

Solve the following equation system: $$x+y+xy=19$$ $$y+z+yz=11$$ $$z+x+zx=14$$ I've tried substituting, adding, subtracting, multiplying... Nothing works. Could anyone drop me a few hints without ...
1
vote
2answers
42 views

Solve the given system of equations

System of equations is: x1 + 2x2 - 3x3 = 9 2x1 - x2 + x3 = 0 4x1 - x2 + x3 = 4 I start by creating the augmented matrix for this system: ...
2
votes
1answer
30 views

Solving simultaneous equations involving complex numbers

This is a question from the book Discrete Mathematical Structures by Bernard Kolman, Robert C. Busby and Sharon Cutler Ross. I want to find the explicit formula of the recurrence relation $g_n = ...
0
votes
1answer
62 views

Two equations with $2n$ variables

Say I have probabilities $p_i$ and $q_i$, $i=1,\dots, n$ and the following two equations \begin{align}\sum _{i=1}^{n}(-p_{i}^{4} +2p_{i}^{3} -2p_{i}^{2} +p_{i} )&=\sum _{i=1}^{n}(-q_{i}^{4} ...
2
votes
1answer
38 views

System of Equations , Finding value of coefficient so equations are inconsistent

Question; Calculate a value for the coefficient $'a'$ of $x$ so that the solutions to the three equations are inconsistent. Demonstrate the resulting system of equations are then inconsistent: ...
0
votes
4answers
144 views

Are there real solutions to $x^y = y^x = 3$ where $y \neq x$?

I need to solve the following equation for (x,y) $$x^y = y^x = 3$$ Everytime I run a numerical method for this problem, I get $$ (x,y) = (1.82546...,1.82546..) $$ I expect there to be a solution ...
1
vote
2answers
53 views

Boolean equation

$$\text{Solve for}\space{x, y}$$ $${a_1, a_2, a_3, a_4, b_1, b_2} \; \text{ - variables}$$ $$\left\{ \begin{aligned} {a_1}\&x \oplus {a_2}\&y &= {b_1} \\ {a_3}\&x \oplus {a_4}\&y ...
3
votes
3answers
54 views

Solving simultaneous trigonometric equations

How do I solve the following trigonometric equations for $\alpha$ and $\beta$: $$ x = d\cos(\alpha + \beta)+h\cos(\alpha) $$ $$ y = d\sin(\alpha + \beta)+h\sin(\alpha) $$ My attempt: Use: $$ ...
1
vote
1answer
27 views

Can I find these parameters to make this fourth degree polynomial has a single real root?

The polynomial is defined as: $$ P(X)=\left(1+\left(\alpha+X\right)^2\right)\left(1+\left(\beta-X\right)^2\right)-\Delta. $$ Can I find positive $\alpha$, $\beta$ and $\Delta$ to make $P(X)=0\iff ...
0
votes
1answer
48 views

possible real solutions of the equations

What are the possible real solutions of the equations $$1000=v_1^2+4v_2^2,100=v_1+4v_2$$ Its a physics question but I thought its not necessary to post here . Thank you.
2
votes
4answers
36 views

Can Extraneous Roots be Introduced by Elimination?

Suppose you have two equtaions: $$2xy + y^2 = 0$$ $$x^2 + 2xy + 1 = 0$$ Subtracting the second from the first yields $y^2 - x^2 - 1 = 0$. Isolating y, we discover that $y = \pm\sqrt{x^2 + 1}$. ...
1
vote
1answer
28 views

Proof of $\sum_{k=1}^n z_k = n$ iff $z_i = 1$ for $|z_k| = 1$ and $z_k \in \mathbb C$.

If we have $n$ complex number $z_k, k = 1,\ldots, n$ with $|z_k| = 1$, then it is intuitively obvious to see that $$ z_1 + z_2 + \ldots + z_n = n \quad\mbox{iff}\quad z_1 = z_2 = \ldots = z_n = 1. ...
0
votes
0answers
35 views

Is this polynomial minimization equivalent to my original problem?

I'm trying to find two vectors $\mathbf{x}$ and $\mathbf{y}$, such that their entries satisfy a system of equations, each one in the form $$\sum_{(i,j)\in J} x_iy_j=0$$ for a given collection of $J$'s ...
7
votes
4answers
735 views

Determining a matrix from its characteristic polynomial

Let $A\in\mathcal{M}_{n}(K)$, where $K$ is a field. Then, we can obtain the characteristic polynomial of $A$ by simply taking $p(\lambda)=\det(A-\lambda I_n)$, which give us something like ...
4
votes
1answer
81 views

obtaining Bernoulli numbers from determinant

I am reading a paper entitled Bernoulli Numbers Via Determinants by Hongwei Chen and I'm confused about a particular step. The author sets up a system of equations via the following: first let $B_n$ ...
-1
votes
1answer
48 views

Linear Algebra 1 equation with 3 variables

The question: solve the equation $8x + 3y - 7z = 12 \\ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -\dfrac 3 8 \\ -\dfrac 8 3 \\ \dfrac 8 7 \end{bmatrix} + s \begin{bmatrix} ...
0
votes
1answer
28 views

How can I solve this system (through approximate or exact methods)?

I have a 4 equations 4 unknowns system as i attached it. I want to calculate T1, T2, P1 and P2 through solve this system and i have coded it in mathematica but it could not obtain the answer by solve ...
1
vote
0answers
17 views

Finding a cubic function with one real root given its graph.

When given a cubic graph with one real root. I need to find the equation of that graph using the function $$y=a(x-s)(x^2+bx+c)$$ where a, b,and c are unknowns. The y intercept is therefore $t = -asc$ ...
0
votes
2answers
33 views

Solve system of linear equations in RREF

I need to solve a system of linear equations using rref. I need to put my answers in the following format: I am assuming that they are two vectors, which one has a scalar s. Could you help me out ...
0
votes
2answers
45 views

Solving a system of equations with variables in denominator.

Solve for $\{x,y,z\}$: \begin{cases}\dfrac1x+\dfrac2y-\dfrac1z=\dfrac43\\\\ \dfrac2x+\dfrac3y-\dfrac2z=\dfrac53\\\\ \dfrac3x+\dfrac4y-\dfrac6z=3 \end{cases} My attempt: I have tried combining ...
1
vote
1answer
47 views

System of trignometric equations.

I'm trying to solve the following system: $$K \sin(s-t) = \sin(s)$$ $$K \sin(t-s) = \sin(t)$$ I've tried playing around with it and learned that $\sin(t) = - \sin(s)$, so $s = -t + 2 n \pi$ for all ...
1
vote
1answer
46 views

Can maple solve a nonlinear system of equations?

I have a system of equations, see below. I wonder if it is possible to solve this for three unknowns $x,y, \gamma$ (the $c_i$ are known constants) in a computer program for ex. maple? I do not wanna ...
0
votes
2answers
63 views

solving system of equations with constant $i$ [closed]

What are the values of $a,b,c$ given the system of equations given below: $a+b+ab=i$ $b+c+bc=2i$ $c+a+ac=3i$
0
votes
1answer
17 views

Solution space for quadratic equations with nilpotent matrices

Let ${\bf w}\in\mathbb{R}^3$ and ${\bf N}\in\mathbb{R}^{3\times 3}$ be a nilpotent matrix with degree 3. Consider the following system of quadratic equations, $$ \begin{align} {\bf w}^\top{\bf w} ...
0
votes
1answer
47 views

Need some help with applying specific boundary conditions to b-spline system of equations

I'm building a package for B-spline interpolation in Julia, and I've come across a boundary condition that I want to implement but can't wrap my head around how to do it (mathematically). Basically, ...
3
votes
2answers
46 views

Geometric progression, two equations problem

We have two equations: $$1. \ a_1 + a_2 + a_3 = 21$$ $$2. \ a_1^2 + a_2^2 + a_3^2 = 189$$ Answer should by $a_1$, $a_2$ and $a_3$. How the title says, these 3 elements are part of geometric ...
0
votes
0answers
22 views

Solution to system of multivariate quadratic equations

I aim to find a vector ${\bf w}\in\mathbb{R}^N$ such that it solves the following system of quadratic equations, $$\forall i,j\quad {\bf w}^\top {\bf A}^{ij}{\bf w} = B_{ij}$$ where ${\bf ...
-1
votes
0answers
49 views

How to solve the Probability Markov chain system of equations

I have this system of equations from a 2-D Markov chain (see the figure. How can i calculate the coefficient matrix, state probability vector and the constant vector from this system of equations. ...
2
votes
2answers
57 views

How do I deal with a floor function is a system of equations?

How would one solve an equation with a floor function in it: \begin{cases} y=12(x-\lfloor x \rfloor) \\ x=12(y-\lfloor y \rfloor) \end{cases} Maybe an algebraic method could be used?
1
vote
2answers
58 views

Solve equation of inverse functions

I have two different functions $y_1=f_1(x)$ and $y_2=f_2(x)$, both invertible but quite complex. I am able to find their inverse functions numerically, i.e. $f^{-1}_1(x)$ and $f^{-1}_2(x)$, by ...
0
votes
1answer
21 views

systems of equations with variables

I have the following problem in my homework Suppose a, b, are two constant paramaters such that the system below is consistent for any values of f and g. What can you say about the numbers a ...
2
votes
1answer
38 views

why does matlab give me a negative number?

I have the following problem A steel company has four different types of scrap metal (called Typ-1 to Typ-4) with the following compositions per unit of volume They need to determine the volumes ...
0
votes
2answers
37 views

System of linear equations - Resolution

$$ \left( \begin{matrix} \pi_1 & \pi_2 & \pi_3 \end{matrix} \right) = \left( \begin{matrix} \pi_1 & \pi_2 & \pi_3 \end{matrix} \right) \begin{bmatrix} 0.6 & 0.3 & ...
0
votes
2answers
28 views

Finding the general solution of a system of linear equations

so I've come across this question in preparation for an exam: Let $A$ be a $4\times 4$ matrix where $rank(A)=3$. The vectors $(1,2,0,-1),(0,2,1,1)$ are solutions to the system ...
0
votes
0answers
22 views

Parametric solutions to underdetermined system of nonlinear (product) equations

I have a system of equations where the left hand side is a constant and the right hand side is a product of some subset of $n$ variables. $$c=\prod_{i\in K} x_{i}$$ Where $K$ is some subset of the ...
0
votes
1answer
52 views

Singular solutions of a system of nonlinear 2nd order ODEs

I'm faced with the following nonlinear 2nd order system of ODEs: $$ \phi''(r)+\frac{4r^3-1}{r^4-r}\phi'(r)+\frac{r^2 h(r)^2+2r(r^3-1)}{(r^3-1)^2}\phi(r)=0, \\ ...