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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23 views

Solve system of polynomials

I have four polynomials with four unknowns $x_{1},x_{2},y_{1}$ and $y_{2}$ as following $$ \left\{ \begin{array}{c} m_{1} + m_{2}x_{1} + (m_{5} + m_{6}x_{2})y_{1} + (m_{3}+m_{4}x_{1})y_{2} ==0 \\ n_{...
0
votes
0answers
28 views

Hadamard product and linear systems

Given two matrices $A, B \in \mathbb{R}^{n \times m}$, where $A = \{a_{i,j}\}$ and $B = \{b_{i,j}\}$, the Hadamard product (or point-wise product) is a matrix $$C = A \circ B$$ such that $C = \{c_{i,j}...
2
votes
1answer
44 views

System of linear equations - geometrical representation of solution

So, if we're given the system: $$x+ay=1$$ $$-ay+z=a-1$$ $$2z=a$$ where $a\neq 0$, we can write it's solutions as: $$x=\frac{a}{2}$$ $$y=-\frac{1}{2}+\frac{1}{a}$$ $$z=\frac{a}{2}$$ or: $$(x,y,z)=\bigg\...
0
votes
0answers
17 views

Existence of solution for nonlinear (algebraic) equations.

Let $f_1(x_1,\cdots,x_n)=0$, $\vdots$ $f_n(x_1,\cdots,x_n)=0,$ be a nonlinear equation. Is there a condition on $f_1,\cdots,f_n$ under which this equation has a solution? Thanks for your help.
1
vote
1answer
66 views

First integrals for solving system of ODEs

Assume a problem $$ \begin{cases} \frac{\mathrm{dx}}{\mathrm{dt}} = \frac{y}{x-y}, \\[2ex] \frac{\mathrm{dy}}{\mathrm{dt}} = \frac{x}{x-y}. \end{cases}$$ Additionally, $x = x(t)$ and $y=y(t)$. ...
2
votes
2answers
50 views

Using augmented matrices to find a number

There's this system of equations $$(8 − a)x_1 + 2x_2 + 3x_3 + ax_4 = 2$$ $$x_1 + (9 − a)x_2 + 4x_3 + ax_4 = 1$$ $$x_1 + 2x_2 + (10 − a)x_3 + ax_4 = 2$$ $$x_1 + 2x_2 + 3x_3 + ax_4 = 2$$ Now I have ...
1
vote
1answer
29 views

Showing that a system of equation is inconsistent

When working out this system of equations, I've found that there are no solutions. Is there any way from the start you can Identify that this system of equations has no solutions? $2x + y − z + u = ...
0
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0answers
51 views

System of equations for semi-unitary matrix

I have a semi unitary matrix $A_{i,j}$ with $1 \leq j \leq N$, $1 \leq i \leq M$ and $M\geq N$, i.e. $A^\dagger A = I$. I now have a set $N$ equations for the squared entries of each row: $$\sum_j |...
1
vote
2answers
33 views

A system of polynomial equations of degree $2$ in two variables

I need to find an explicit solution of this system of polynomial equations of degree $2$ in two variables $x,\,y$: $$\begin{cases} p_1x^2+q_1y^2+r_1xy+s_1x+t_1y+u_1=0\\ p_2x^2+q_2y^2+r_2xy+s_2x+t_2y+...
0
votes
2answers
28 views

How to numerically find zeros of a system of first-order differential equations (Airy function)?

To numerically approximate the Airy function y = Ai(x) which satisfies the equation $$ y'' - xy = 0 $$ I converted this second-order diff. eq. into a pair of first order diff eq. and solved them using ...
2
votes
2answers
53 views

Systems of equations with unknown constant

How do I solve this system? It says I must row reduce it to solve it (depending on parameter $a$). $(8−a)x_1 + 2x_2 + 3x_3 + ax_4 = 2$ $x_1 + (9−a)x_2 + 4x_3 + ax_4 = 1$ $x_1 + 2x_2 + (10−a)x_3 + ...
1
vote
1answer
39 views

Solve for two variable in terms of other [closed]

How do I solve for $\alpha$ and $\beta$ in terms of $\theta$ using the equations $$a^2 \cos^2\theta \:+\:b^2 \sin^2\theta \:=\:a^2\cos^2\alpha $$ and $$b^2 \cos^2\theta \:+\:a^2 \sin^2\theta \:=\:a^2\...
2
votes
0answers
36 views

Derivative solution of $\frac{(x\cos\,\theta-y\sin\,\theta)^2}{a^2}+\frac{(x\sin\,\theta+y\cos\,\theta)^2}{b^2}=1$

The equation $$\frac{(x\cos\,\theta-y\sin\,\theta)^2}{a^2}+\frac{(x\sin\,\theta+y\cos\,\theta)^2}{b^2}=1 \ \ \ \ \ \ \ \ (*)$$ has the following expression for the derivative: $$\frac{\mathrm dy}{\...
0
votes
3answers
31 views

Algebraic system of equations problem

Solve the follow system of equations: $$x+y+z=5$$ $$\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}=5$$ $$x^3+y^3+z^3=53$$ Thanks for any help.
2
votes
1answer
18 views

Chinese Remainder Theorem When GCD is not 1

I've got this system of equations that I'm trying to solve. I'm pretty sure I have to use the CRT, but to my understanding, it can only be used when GCD of all the mods is 1. I don't want an answer ...
1
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2answers
26 views

normalization of constraints $ 0 \leq x \leq 1 $ in Lagrangian KKT

With Lagrangian we have an objective function and a set of equality constraints of form $ g_{i}(x_{j}) = 0 $ . With KKT we can have another set of inequality constraints of the form $ h_{i}(x_{j}) \...
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votes
1answer
20 views

System of congruences of 2 unknowns

Given constants $A, B, C, D$, and unknowns $x, m$, how would I go about solving a system such as this: $$A\equiv x B\mod m$$ $$C\equiv x D\mod m$$ I'm certain there is a very simple equivalent form of ...
3
votes
4answers
92 views

I'm stuck in a logarithm question: $4^{y+3x} = 64$ and $\log_x(x+12)- 3 \log_x4= -1$

If $4^{y+3x} = 64$ and $\log_x(x+12)- 3 \log_x4= -1$ so $x + 2y= ?$ I've tried this far, and I'm stuck $$\begin{align}4^{y+3x}&= 64 \\ 4^{y+3x} &= 4^3 \\ y+3x &= 3 \end{align}$$ $$\begin{...
1
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1answer
92 views

Solve the system of equations $x^y=y^x$

Solve the system of equations $$ x^y=y^x \\ a^x=b^y $$ I could not solve this despite many tries
0
votes
2answers
33 views

number of solutions of system of two equations, two unknowns (Matrix)

How can we find that when a system of two equations, two unknowns has Infinite Solutions. I want a solution with matrix. I know this method (which is not with matrix): $ax + by = c$ $a'x+ b'y = c'$ ...
0
votes
0answers
47 views

Solution to System of Complicated Differential Equations

I'm looking for a solution to this set of complicated differential equations: $$\begin{align} \dfrac{dθ}{ds} & = \dfrac{\cos θ}{r} − z\\ \dfrac{dz}{ds}& = − \cos θ \\ \dfrac{dr}{ds} &= \...
1
vote
1answer
19 views

How to determine if an object in space is pointing at (oriented toward) another object?

QUESTION: You know the position of two objects in space (one also has an orientation). How do you determine when the object is pointing/oriented at the other object? Hopefully this question makes ...
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votes
2answers
61 views

Simple but hard 2 by 2 system in $x$ and $y$ [duplicate]

Is there a systematic way of solving this system, analytically? $$\begin{cases} x \ + \ y^2=11\\ x^2+y\ \ =\ 7\\ \end{cases} $$ I mean, other than brute-force.
0
votes
0answers
29 views

Probabalistic solve of system of equations

I'm engineer, not mathematician, so excuse me for wrong terminology, but I hope you'll understand the problem. Example situation: I have N electronic components. Each of them has reactance and ...
3
votes
3answers
39 views

Maximize system of linear equations

Suppose you have the system $$ \begin{bmatrix} 4 & 3\\ 1 & 7\\ 5 & 9\\ 2 & 4\\ \end{bmatrix} \begin{bmatrix} x\\y\end{bmatrix}=\begin{bmatrix}b_1\\b_2\\b_3\\b_4\end{bmatrix} $$ How ...
0
votes
0answers
23 views

Solve the EDO $p'={\alpha}p^a+{\beta}p^b,\quad t>0,$

Fix $\alpha , \beta \in (0,\infty)$ . Use Osgood's criterion to show that the equation $$p'={\alpha}p^a+{\beta}p^b,\quad t>0,$$ has at most one nonnegative solution if $a,b \ge 1$. Also, prove ...
0
votes
1answer
18 views

Value of $a$ if system of equation is consistent.

If the following equations are consistent and have more than one solution, what is the value of $a$? Given $u+v=-(av+1)$ $u+2v=-a(v-1)$ $3u+8v=a+2$ I was thinking that system of equation is ...
0
votes
0answers
43 views

Linear system equations

I need to get, preferably by a numerical method, a solutions of: $$\left\{\begin{array}{lll} 2\sum_{i=1}^n b_{ik}x_i+x_{n+1}=0&\text{for}& k=1,2\ldots,n\\ \sum_{i=1}^n x_i=0 \end{array}\right.$...
0
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0answers
33 views

Solving system of nonlinear equations via iteration

I will give an example to illustrate the question: Assume I have the system: $$ xy + x + y = 7\\ x^2 + y^3 = 9 $$ and I want to solve for $x$ and $y$. It is a fairly common approach to rearrange ...
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2answers
35 views

Decouple a system of two second order differential equations

I have a system of second-order differential equations that I want to decouple. they are, $\ddot{x} = \frac{\omega_1^2}{2} x + \omega_2 \dot{y}$ and $\ddot{y} = \frac{\omega_1^2}{2} y - \omega_2 \...
1
vote
3answers
38 views

Process for solving this system of equations

I have this system of equations for which I'd like to solve for $x$,$y$, and $r$ where $a$,$b$, and $t$ are constants: 1: $0 = (x-a)^2 + (y - b)^2 - t^2$ 2: $y = \dfrac{bx-rx+ar}{a}$ 3: $r = \dfrac{...
0
votes
1answer
37 views

How to solve simultaneous inequalities (reasked)? [duplicate]

I am doing multivariable calculus, and specifically double integrals. I am facing difficulties finding the domain of the integal, however i am given the following equations: $$1≤2x+y≤2$$ $$0≤x−2y≤1$$ ...
-2
votes
1answer
45 views

Find an energy functional for the nonlinear viscous oscillator $x' = v$, $v' =-b(v)v-k(x)x$, $t>0$ [closed]

Consider the nonlinear viscous oscillator $$\begin{cases} x' = v\\ v' =-b(v)v-k(x)x,\quad t>0, \\ \end{cases}$$ where $(x,v)$ is the position and velocity of the oscillator. Here $b : \mathbb R\...
0
votes
1answer
21 views

Showing that a system has a unique steady state at $(0, 0).$

Consider a system: $$ dx/dt = y + x(2 − x^2 − y^2 ), $$ $$dy/dt = −x + y(1 − x^2 − y^2)$$ (i) Show that the system has a unique steady state at $(0, 0).$ My immediate thought is to simply ...
0
votes
0answers
9 views

Hyperplane and cubic curves and their intersections.

Solve the following: $$a^4-a^2+A_{1}E u=0;$$ $$b^4-b^2+A_{2}Eu=0;$$ $$c^4-c^2+A_{3}Eu=0;$$ $$d^4-d^2+A_{4}Eu=0;$$ and $$a^2+b^2+c^2+d^2=E,$$ for $a, b, c, d,$ and $u$, when $A_{1}, A_{2}, A_{3}, A_{4}...
0
votes
1answer
29 views

Question regarding systems of equations

If I have the following system of equations: $2+x^2-y^2=0$ $x^2-y^2-2=0$ And if I substitute $y$ by a function of $x$ and vice versa I get: $2+x^2-x^2+2=0$ $y^2-y^2-4=0$ I therefore get: ...
0
votes
0answers
20 views

Linear algebra: Solving for the coefficients on vectors

I am solving the following system: $$ -\frac{1}{r^2}\begin{bmatrix}\sqrt{\mu}\cos(\theta)\\ \sin(\theta) \end{bmatrix}= (v_r'-v_{\theta}\theta')\begin{bmatrix}\frac{\cos(\theta)}{\sqrt{\mu}...
0
votes
2answers
47 views

Find the eigenvector and eigenvalues for the following 3 x 3 Matrix?

$$ \pmatrix{5 & 8 & 16 \\ 4 & 1 & 8 \\ -4 &-4 & -11} $$ I already got the eigenvalues that is $\lambda = 1$ and $-3$. And I managed to solve the eigenvector corresponding to ...
3
votes
5answers
64 views

Solution of $x^y=y^x$ and $x^2=y^3$

Solve the given set of equations: $x^y=y^x$ and $x^2=y^3$ where $x,y \in \mathbb{R}$ Would any other solution exist other that $x=y=1$ because I think $x^2=y^3$ will only be true for $x=y=1$ or $x=y=...
0
votes
1answer
21 views

Point of intersection of ellipses

If two ellipses are intersecting at a point,is it necessary that the line drawn joining the centre of those two ellipses should also pass through the point of intersection (of ellipse)? (if yes,how to ...
0
votes
0answers
18 views

Solving systems of linear equations with complex numbers by hand

How can I solve a 3x3 system of linear equations with complex numbers by hand without making a mistake? I know that I can solve them either with Gaussian Elimination or Cramer's rule, but I find it ...
1
vote
1answer
48 views

How to estimate the parameters of a logistic differential equation from the values of its solution at times 0, 1 and 2?

How do I solve this system of equations? I received these equations after letting Wolfram Alpha solve the logistic differential equation $$N'(t)=kN(t)(M-N(t)),\qquad N(0)=65,$$ that outputs: $$N(t)=\...
0
votes
1answer
41 views

Solve $Ax=b$ for $A$ in MATLAB

I have this linear system $$\begin{bmatrix} cY(t-1)\\ acY(t-1) - acY(t-2)\end{bmatrix} = T \begin{bmatrix} Y(t-2)\\ Y(t-1)\end{bmatrix}$$ where both $c$ and $a$ are known constants, and I need to ...
0
votes
2answers
81 views

Solve the simultaneous equations for real numbers $x$ and $y$: $ \sqrt{x+a} + \sqrt{x-a} = 3 $ and $ x+y=5 $

Question: Let $a$ be a real number. Solve the simultaneous equations for real numbers $x$ and $y$: $$ \sqrt{x+a} + \sqrt{x-a} = 3 $$ $$ x+y=5 $$ My attempt: Consider the ...
2
votes
0answers
38 views

Solving a 1D integral with system of equations for retarded electromagnetic fields

I need to solve the following integral to calculate the effect of retarded electromagnetic fields on a test charge: $\int\limits_0^\zeta\frac{(\psi-(1+x)\sin(\psi+\alpha))(\frac{\psi^2}{2\beta^2(1+x)}...
1
vote
2answers
43 views

Solving a system of equations which contain sin and cosine terms.

Hello my question is the following: Solve the given system of equations: $$E=\frac{l_{p}}{\pi}\sqrt{\sin^{2}\left(\frac{\pi y_{1}}{l_{p}}\right)+\sin^{2}\left(\frac{\pi y_{2}}{l_{p}}\right)+\sin^{2}\...
0
votes
1answer
74 views

When I know $a+b+c, a^2+a^2+b^2, a^3+b^3+c^3$, then how can I find the $a$ and $b$ and $ c$ [closed]

When I know $$a+b+c = A$$ $$a^2+a^2+b^2 = B $$ $$a^3+b^3+c^3 = C$$ Then how can I find the $a$ and $b$ and $c$?
0
votes
1answer
30 views

Solutions to set of equations involving prime numbers

Is there a collection of distinct positive integers $(k_1, k_2, k_3, p_1, p_2, p_3)$ such that: $p_1, p_2, p_3$ are odd primes, and $k_1, k_2, k_3$ are odd $(k_1 + 2) p_1 = k_2 p_2$ and $(k_2 + 2) ...
1
vote
0answers
23 views

Transform the system of trigonometric equations

How to extract $\ell$ and $L$ from the following system of equations: $$\alpha=\arctan {R_E \cos \ell \sin L \over R_0 + R_E(1 - \cos \ell \cos L) }$$ $$\beta=\arctan {R_E \sin \ell \cos \alpha \over ...
4
votes
2answers
84 views

How to solve simultaneous inequalities?

I am doing multivariable calculus, and specifically double integrals. I am facing difficulties finding the domain of the integal, however i am given the following equations: $$1 ≤ 2x+y ≤ 2$$ $$0 ≤ x-...