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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1answer
44 views

Changing variables for a partial differential equation

If I have the following systems of PDE \begin{align} 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, ...
1
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1answer
51 views

Solve non-linear ode system as a function of $t$.

I need to solve this ode' system $$ \begin{cases} \dot x=y\\\dot y=-x+x^2=x(x-1) \end{cases} $$ To solve it as a function $x(y)$ or $y(x)$ is trivial, but I need the solution as a function of time: ...
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1answer
26 views

Notation of a linear inequality system.

Sorry to bother with this rather trivial question, but nowhere in my lectures or books can I quite find out what the topmost line means. Maybe I'm forgetting something. Anyway: Line 2 and 3 are ...
0
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1answer
117 views

Using QR decomposition to solve a system of equations with a singular matrix

If $A\in\mathbb{R}^{n\times n}$ is singular and $x,b\in\mathbb{R}^{n}$ are such that $Ax=b$, am I right in thinking that the upper triangular matrix $R$ of $A$'s $QR$ decomposition must have at least ...
0
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1answer
18 views

How should I find the analytical form of these recursive equations

I have $$x_1(t+1) = (1-m \rho_1)x_1(t) + n\rho_2 x_2(t) + h1$$ $$x_2(t+1) = (1-m \rho_2)x_2(t) + n\rho_1 x_1(t) + h2$$ Suppose $x_1(0)$ and $x_2(0)$ are known. How can I find the analytical form of ...
17
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8answers
3k views

Kid's homework: 4 equations 5 unknowns? Going crazy!

I'm new here, and I'm hoping someone can help out. My 10 year old son has been set a maths problem, which I can't solve. I've got a PhD in neuroscience and do a fair amount of matlab stuff (data ...
0
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1answer
57 views

near identity change of coordinates

Problem: Consider the scalar differential equation $$x' = \frac{4x – 24x^2 – 16x^3}{1 – 12x – 12x^2}.$$ which has a fixed point at $x^* = 0 $. For $x$ close to $x^* = 0 $ find a near identity ...
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0answers
19 views

Insights in solving systems of eqn?

So, I need to find all solutions in integers of the following system: $x_1 + x_2 + 4x_3 +2x_4 =5 $ $-3x_1 - x_2 + 0 - 6x_4 =3$ $-x_1 - x_2 + 2x_3 - 2x_4 =1$ I know the steps, but I don't ...
0
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3answers
39 views

Cant do this system problem

Question: For the value(s) of $k$,if any,will the following system have (a) no solution, (b) a unique solution, (c) infinity many solutons: $$x+y+kz=1,\\x+ky+z=1,\\xk+y+z=-2.$$ answer: for $k=1$ ...
0
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1answer
43 views

Solve a system of inequalities

$$\begin{cases} \log_{2}^{2}(-\log_{2}x) + \log_{2}\log_{2}^{2}x \leq 3 & \\-4 |x^2-1|-3\geq \frac{1}{x^2-1}& \end{cases}$$ What I've tried: Make substitution $t=x^2-1$ and solve second ...
3
votes
2answers
95 views

the global stable and unstable manifolds

Show that $x^* = (1, 2)$ is a fixed point of the system $x_1' = 2 + 3x_1 − 2x_2 − x_1^2 + 2x_1x_2 − x_2^2$ $x_2' = 3 + 4x_1 − 3x_2 − x_1^2 + 2x_1x_2 − x_2^2$ Determine $W^s(x)$ and $W^u(x)$, the ...
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3answers
58 views

Solve the System of Equations in $x$ and $y$

\begin{equation} x+\frac{3\,x-y}{x^2+y^2}=3 \tag{1} \end{equation} \begin{equation} y=\frac{x+3\,y}{x^2+y^2} \tag{2} \end{equation}
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1answer
36 views

What is the canonical basis of a dualspace in $\mathbb{R}^3$?

I have the following: Consider the basis $$B := \{\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \}$$ of the ...
0
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3answers
67 views

Choose h and k such that the system has a solution, a unique solution and many solutions.

Im learning linear algebra, and im tasked with choosing $h$ and $k$ such that this system: $$ \begin{cases} x_1+hx_2=2\\ 4x_1+8x_2=k\\ \end{cases} $$ Has (a) no solution, (b) a unique solution, and ...
0
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1answer
127 views

Solve Coupled System of Equations via Matrix

I have a coupled system of three equations that I am trying to solve via matrices and I am having trouble figuring out how to write out my matrices. My three equations are as follows: $-sx+sy=0$ ...
0
votes
1answer
54 views

Can someone please provide an intuition behind cramer's rule?

See question. I usually get concepts like this very quickly (no studying required), but this one looks like Chinese. Can someone please help me understand a brief intuition behind Cramer's rule for ...
0
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4answers
124 views

Solving a logarithmic system of equations

I am working on a test study guide and I can't seem to get the correct answer for this system of equations: \begin{align*} \ln(x) &= 3\ln(y) \\ \ 3^x &= 27^y \end{align*} I'm not ...
0
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2answers
31 views

what are some typical systems of equations generating from practical problems?

I want to know some typical forms of system of equations generating from practical problems in engineering/economics/physics,etc. Some examples or research articles would be good. Specifically, I am ...
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2answers
18 views

$3$ lines $4$ variables linear equation gaussian

So I'm currently taking a Linear Algebra class and am stuck on a problem. I have the equations: $$\begin{cases}\begin{align}&x + 2y - z + 3t = 3\\ &2x + 4y + 4z + 3t = 9\\ &3x + 6y - z ...
5
votes
2answers
128 views

How find this system $a^2+b^2=3,a^2+c^2+ac=4,b^2+c^2+\sqrt{3}bc=7$

Find the this system real solution $$\begin{cases} a^2+b^2=3\\ a^2+c^2+ac=4\\ b^2+c^2+\sqrt{3}bc=7 \end{cases}$$ I think that one can use Geometry to solve this system. Maybe there exist an ...
2
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2answers
59 views

System of differential equations using substitution

Exact problem statement Solve the system $\left\{\begin{matrix} x_{1}'(t)=3x_{1}(t)-2x_{2}(t)+e^{2t},x_{1}(0)=a & \\ x_{2}'(t)=4x_{1}(t)-3x_{2}(t),x_{2}(0)=b & \end{matrix}\right.$ by using ...
2
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3answers
60 views

Find all solutions to this system of congruences

$$x \equiv 11 \pmod{84} $$ $$ x \equiv 23 \pmod{36}$$ I have the bulk of the work done for this; $x=11+84j$ $x=23+36k$ $\Rightarrow 11+84j \equiv 23 \pmod{36}$ $\Rightarrow 84j \equiv 12 \pmod{36}$ ...
1
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2answers
66 views

Find an integer $x$ satisfying the congruence:

$$x \equiv \ 1 \pmod3$$ $$x \equiv \ 2 \pmod5$$ $$x \equiv \ 8 \pmod{11}$$ From the first, I have $x=3k+1$, $x=5j+2$ from the second and $x=11l+8$ from the third. Subbing the third into the second I ...
2
votes
2answers
57 views

Describe all integers a for which the following system of congruences (with one unknown x) has integer solutions:

$$x\equiv a \pmod {100}$$ $$x\equiv a^2 \pmod {35}$$ $$x\equiv 3a-2 \pmod {49}$$ I'm trying to solve this system of congruences, but I'm only familiar with a method for solving when the mods are ...
1
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3answers
123 views

Solving a nonlinear system of equations.

Given that $x,y,z\in\mathbb R$, solve $$\begin{cases}6x^2-12x=y^3\\6y^2-12y=z^3\\6z^2-12z=x^3\end{cases}$$ I've tried adding the equalities but to no avail. I'd add what I've tried, but it'd be ...
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1answer
31 views

Methods for solving linear systems

This is such a basic topic but there are so many different methods proposed for solving a linear system of equations. I recently found a very good source but couldn't really make sense of all the ...
1
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3answers
135 views

Creating and solving large systems of equations

I am trying to follow a solution in a book so that I can build my own model. They produce the set of equations below. The book claims it to be a system of equations with 10 unknowns; however from my ...
0
votes
1answer
23 views

Systems of Linear equations (substitution method) *Got Part A*IDK about Part B*Part C i have no clue?*

Part A: write the equation that represents M................. y=2x-5 Write the equation that represents N.............................. y=3x+2 ( that is right^^^^^) Part B:using the equations you ...
0
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1answer
156 views

How to solve an overdetermined linear system given equations with different uncertainties

Please, I would like some help to solve the following problem: I have an overdetemined system of linear equation and want to minimize overall error. Up to now, not a problem, I could use least ...
0
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1answer
14 views

If M = S, how to isolate a?

So I have to isolate $a$ in $M=S$ $M=1+\dfrac{a}{b}$ $S=a+b$ So, I put it up like this: $1+\dfrac{a}{b}=a+b$ ... right? But then what?
0
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1answer
36 views

Periodic system

We have the following system: $\dot{x}=x-y-x(x^2+y^2)$ $\dot{y}=y+x-y(x^2+y^2)$ Determine the equilibrium points Show that this system has a periodic solution. Use the following substitution ...
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2answers
95 views

A system of nonlinear differential equations

We have the following system in $\mathbb{R}^{2}$ $$\dot{y}_1=2-y_1y_2-y_2^2$$ $$\dot{y}_2=2-y_1^2-y_1y_2$$ i) Calculate the equilibrium points en determine their stability. ii) Draw the Phase ...
1
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3answers
36 views

Orthogonal vectors and linear systems

Let us suppose we want to solve, with respect to x, the following equation $\mathbf{a}^\intercal\mathbf{b}\;x = 0$ where $\mathbf{a}, \mathbf{b} \in \mathbb{R}^{n} \setminus \{ 0 \}$. It seems clear ...
2
votes
2answers
51 views

Determining the necessary values for a matrix' coefficients to achieve a certain rank.

I'm having a headache with this... Given the augmented linear system matrix: $$A = \begin {cases} 1 & 0 & 0 & 2 \\ 0 & a-2 & 0 & 0 \\ 0 & 0 & b + 1 & c \\ 0 ...
5
votes
2answers
254 views

Complex numbers system of equations problem with 5 variables

Let $z_0$,$z_1$,$z_2$,$z_3$ and $z_4$ such that $z_i\in C$ that hold: $$(1)|z_0|=|z_1|=|z_2|=|z_3|=|z_4|=1$$ $$(2)z_0+z_1+z_2+z_3+z_4=0$$ $$(3) z_0z_1+ z_1z_2+z_2z_3+z_3z_4+z_4z_0=0$$ Prove that ...
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0answers
110 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
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1answer
399 views

Find the values of $k$ that make this system inconsistent, with unique solution, and with infinite solutions.

I've learned to find the solutions to linear systems using Gaussian Elimination. Moving on, I've found a new kind of exercise I hadn't done before: Find the values for $k$ that make this system: ...
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2answers
43 views

What did I do wrong with Gaussan Elimination for $\begin {cases} x + 5y + 11z = -5\\ 2x + 3y + 8z = 4\\ -x + 2y + 3z = -9 \end {cases}$?

Having problems with this one using Gaussian Elimination. Find the solutions for the linear equation system: $$\begin {cases} x + 5y + 11z = -5\\ 2x + 3y + 8z = 4\\ -x + 2y + 3z = -9 \end ...
0
votes
1answer
70 views

Differential Equations - Method of Undetermined Coefficients for products of polynomials and sines

Consider $y''+y= 2x \sin (x)$ I have the solution for the homogeneous equation. Now i am trying to guess a particular solution for: $2x \sin (x)$ My first guess was: $(Ax+B) \cos x + (Cx +D) \sin ...
4
votes
3answers
58 views

Invalid subtraction when solving system of equations?

I'm trying to solve these two equations: $$\begin{cases} 1-4x(x^2+y^2)=0 \\ 1-4y(x^2+y^2)=0 \end{cases}$$ and I tried to do it by subtracting the first equation from the second, yielding ...
0
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0answers
44 views

Solving equation for powers

I would like to find $\gamma$ in: $$ \sum_{i=0}^n x_i^\gamma = y $$ where $n$, $0 \leq x_i \leq 1$ and $0 \leq y \leq n$ are known. Also, $n$ can be fairly large (i.e. from a few thousands to a few ...
0
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5answers
48 views

How to solve this homogeneous system, with a missing column?

Find the solution set of triplets $(x,y,z)$ that fulfil this system using Gauss-Jordan: $$\begin {cases} -x + 2z = 0\\ 3x - 6z = 0\\2x - 4z = 0\end {cases}$$ First of all, I don't see any ...
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0answers
42 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
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4answers
70 views

Are there no solutions for $\begin {cases} 2x+4y = 6\\ 3x+6y = 5\end {cases}$?

I'm trying to solve an equation system using Gauss-Jordan. $$\begin {cases} 2x+4y = 6\\ 3x+6y = 5\end {cases}$$ So, first, the augmented matrix: \begin{bmatrix} 2&4&5\\ 3&6&6\\ ...
2
votes
1answer
138 views

Can all equation systems be reduced to the identity matrix?

I'm trying to learn about solving equation systems using the Gauss-Jordan method. So, you have to convert the equation system to a matrix, and then reduce it to the identity. When you transform it to ...
0
votes
1answer
38 views

Intersection of linear and quadratic functions

I've been stuck on some math work and I'm not sure how to do it. It involves finding the point where a quadratic and linear function intersect only once. Determine the value of $k$ such that $g(x) = ...
1
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2answers
80 views

linear differential equation problem [closed]

Consider the following system of linear differential equations: $$\begin{split} \frac{dx}{dt}&=−3x+y\\ \frac{dy}{dt}&=x−3y \end{split}$$ Find the eigenvalues and eigenvectors associated ...
5
votes
2answers
72 views

Number of solutions for a system of polynomial equations

Consider the given system of polynomial equations, where all the coefficients are in $\mathbb{C}$: $$\begin{cases} y^n=P(x)\\ Q(x,y)=0\end{cases}$$ I would like to establish that either this system ...
0
votes
0answers
30 views

perturbation solution of two singular ODEs

I need some help with solving the following system of ODEs: $$\epsilon \frac{dx}{dt}=Ay +ABx(1-y)$$ $$\epsilon\frac{dy}{dt}=Bx(1-y)-y-\epsilon y$$ I'm confused by the fact that both equation are ...
1
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0answers
89 views

Linearization of coupled differential equations

Consider the following coupled differential equation: $\dfrac{\partial x(t)}{\partial t}=(a+y(t))x(t)+b(m_1(t)+m_2(t))y(t)$ $\dfrac{\partial y(t)}{\partial t}=cy(t)+d(m_1(t)+m_2(t))x(t)$ $a,b,c,d$ ...