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
37 views

solving a linearly-constrained sparse linear least-squares problem

Given the system of equations $Ax=b$, subject to $Cx\le d$ where $A$ is an $n\times m$ matrix (with $n>m$) and is very large and sparse. As an example $A$ can have $3126250\times 2740$ elements. ...
4
votes
1answer
66 views

Due to numerical inaccuracy, the solution of a boundary value problems becomes negative

I treat a toy example to get my point across. In reality I have to deal with a much more complex model. Let us consider a one dimensional boundary value problem using the bvp5c solver in Matlab. Two ...
0
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0answers
36 views

solving simple system of cubic equations

I know, that there is no general answer as how to solve a system of equations. But mine has a pretty special form. Let $x,a,b \in \mathbb{R}^n$ and $a,b$ are known. I want to find the solution of the ...
1
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0answers
47 views

Frobenius Method For System of Differential Equations

I have a system of ODEs. Can you explain how to solve a system of ODEs using the method Frobenius expansions ? There are 5 ODEs which are coupled and 5 variables. $\omega\hat\rho + i\alpha V_z ...
0
votes
2answers
34 views

Solvability of this linear equation system and finding particular solution/

I have been given a task, that involves determining if this lin.eq.system $$ x_1+2x_2-3x_3+10x_4-x_5=7\\x_1-2x_2+3x_3-10x_4+x_5=9\\x_1+6x_2-9x_3+30x_4-3x_5=5 $$ has a solution by using what our ...
0
votes
1answer
32 views

How to Split a 2D Gaussian pdf into a Grid of Equally Sized Volumes

Let $f(x,y)$ be a Gaussian pdf for some known mean and covariance. Given $(x_0, y_0)$ and $(x_N, y_M)$ such that $$\int_{x_0}^{x_N} \int_{y_0}^{y_M} f(x,y) dy dx \approx 1$$ I would like to split ...
0
votes
1answer
34 views

Difficult augmented matrices question.

I'm currently revising for a maths module that I am taking as part of my physics degree. The final part of the matrices section of a paper I was doing included this question: Solve the this set of ...
0
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1answer
47 views

FermiPasta-Ulam problem

Consider $H(q,p) = \frac{1}{2} \sum\limits_{j=1}^{n+1} {(p_j^2 + (q_{j}-q_{j-1})^2)}$ $H(q,p) $ is the Hamiltonian considered in the FermiPasta-Ulam problem. Consider canonical transformation $Q = ...
4
votes
4answers
94 views

Solve system of nonlinear equations using non-numerical method

Is there any non-numerical method to solve this kind of system of nonlinear equations for $c_1, c_2, x_1, x_2$: $$c_1+c_2 = 1$$ $$c_1x_1+c_2x_2 = 1$$ $$c_1x_1^2+c_2x_2^2 = 2$$ $$c_1x_1^3+c_2x_2^3 = ...
2
votes
3answers
53 views

Does adding two linear equations will result in a line which will pass through an intersection of the linear equations?

I was wondering why it is almost impossible to find a geometrical explanation of why adding two linear equations helps us to find a solution of a system of linear equations? Am I right that adding two ...
4
votes
2answers
40 views

Prove or disprove the system about $n$th power has only one solution $x=y=1$

$$\begin{cases}x^n+y^n=2\\x+y=2\end{cases}\;,\;n\in\mathbb{N}\;,\;x,y\in\mathbb{R}\;,\;n>2$$ I have tried to show that $\displaystyle y'=-\frac{x^{n-1}}{y^{n-1}}=-1$ $$......$$ therefore $x=y=1$ ...
1
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1answer
45 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 $\beta$, $\gamma$ ...
0
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0answers
22 views

Counting the roots of nonlinear systems of equations

I have a "nice" function (vector field) $$\mathbf{f}: \mathbb{R}^n \rightarrow \mathbb{R}^n$$ and I need to find how many roots (zeros) it has in a certain domain (hopefully prove that it has at most ...
0
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4answers
84 views

What am I doing wrong here?

Consider this system of equations: $$ \begin{cases} x+y=6\\x-y=5\\2x+3y=7 \end{cases} $$ This is an overdetermined system and doesn't have a solution (the 3 lines don't intersect). But by adding 2nd ...
1
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0answers
44 views

Phase portrait of DS with skew symmetric matrix

How should I draw phase portrait of DS: $x'=Ax$, where $$A=\left( \begin{array}{ccc} 0 & 1 & 0 \\ -1 & 0 & -2 \\ 0 & 2 & 0 \\ \end{array} \right)?$$ Eigenvalues here are $0, ...
1
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0answers
50 views

find $x$, given $\{c_ix = k_i + y_i\}_{i=[1,n]} $

Given $$c_1x = k_1 + y_1 $$ $$c_2x = k_2 + y_2 $$ $$\vdots $$ $$c_nx = k_n + y_n $$ where the values of $\{c_1 \ldots c_n \}$ and $\{ k_1 \ldots k_n \}$ are known, and $x, \{y_1 \ldots y_n \}$ are ...
5
votes
0answers
98 views

How to solve a particular initial-boundary value problem

I have the following initial-boundary value problem $$\begin{cases}\dfrac{\partial^2 u_1}{\partial x^2}=A_{11}\dfrac{\partial u_1}{\partial t}+A_{12}\dfrac{\partial u_2}{\partial ...
3
votes
0answers
47 views

Solving a system of 3 variables

How to solve or what is the algorithm to solve a system of equations like this: $$\eqalign{ (x +\phantom{3} z)^2 + (y +\phantom{3} w)^2 &= 52\cr (x + 3z)^2 + (y + 3w)^2 &= 296\cr (x ...
1
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1answer
39 views

How to solve system of nonlinear differeintial equations

System follows: $$ y'=\frac{y^2}{z-x}; z'=y+1$$ I was found the 2 ways. The both are wrong 1) $$z = x + \frac{y^2}{y'}; z'=1+\frac{2yy'^2-y^2y''}{y'^2}=y+1; => (p(y) = y')=> yp(yp'+p)=0;$$ ...
1
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2answers
70 views

System of Nonhomogeneous DEs - Help Solving???

I'm studying for finals at the moment and could use some help with solving the particular solution for this system of nonhomogeneous differential equations: $x' = \begin{bmatrix}1 & 0\\ 2 & ...
0
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1answer
14 views

How to Do Trilateration?

Trilateration is the process of calculating the coordinates of a point by using its distances to three other points. Say that, we have three points of which we know the coordinates: $A(A_x, A_y)$ ...
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2answers
91 views

Solving the particular solution of system of nonhomogeneous DEs???

I am studying for finals at the moment, and I'm trying to better understand using the method of underdetermined coefficients to solve a system of DEs. Here's an example of one I'm stuck on at the ...
1
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1answer
11 views

Finding values $a$ and $b$ which transforms a differential equation

Given the function $u = u(\xi, \eta)$ where $\xi = x + ay$ and $\eta = x + by$, find the values of $a$ and $b$ such that they transform the equation $$\frac{\partial^{2}u}{\partial x^{2 }} + ...
1
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3answers
28 views

Why the linear numerator for fractions with irreducible denominators and constant numerators for reducible denominators? [duplicate]

For example: $\Large{\frac{2x^3+5x+1}{(x^2+4)(x^2+x+2)}}$ breaks down to $\Large{\frac{ax+b}{x^2+4}+\frac{cx+d}{x^2+x+2}}$ I have been told that since the denominators are irreducible, the ...
0
votes
0answers
14 views

Criterion of removal of equations from overdetermined system

Consider the problem of solving overdetermined system Ax = b; In the problem I am trying to solve (from the field of spectral unmixing) number of unknowns usually varies between N = 2 and 5 and the ...
2
votes
5answers
71 views

Why a linear numerator for fractions with irreducible denominators?

For example: (2x^3+5x+1)/((x^2+4)(x^2+x+2)) breaks down to (ax+b/(x^2+4))+(cx+d/(x^2+x+2)). I have been told that since the denominators are irreducible, the numerators will be either linear or ...
1
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3answers
66 views

Partial fraction decomposition of a complicated rational function

Find the partial fraction decomposition of the rational function $\displaystyle \frac{2x^3+7x+5}{(x^2+x+2)(x^2+1)}$ I have tried dividing first but keep running into problem after problem, please ...
0
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0answers
28 views

determine in what grid rhombus is a point

i have a rhombus ( i.e. diamond) grid determined by these equations ...
1
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1answer
53 views

Solve the system of equations…!

Can you please help me solve this system of equations (frankly I have no idea, it's the first equation of this type that I solve, so please, write only a hint): $$ \left\{ \begin{array}{c} x-\arctan ...
2
votes
3answers
28 views

Solve the following system of linear equations for any values of real parameter $a$…

For any values of parameter $a$ solve the following system of linear equations: $$\begin{cases} x+y+2z=1 \\ 2x+ay-z=4 \\ 3x+y+3z=1 \end{cases} $$ Calculating the value of determinant I found out, ...
0
votes
1answer
73 views

Linear systems. Please help me solve this

Please help me solve this. Consider for every real number $a$ the linear system of equations: $$ \begin{align} x +( a + 1 )y + a^2 z &= a^3 \\ (1-a)x +( 1 - 2a )y &= a^3 \\ x +( a ...
0
votes
1answer
38 views

Understanding a solution of a system of differential equations

I am looking at the system $$ x' = -y + xy^2 \\ y' = 4x - 4x^2 y $$ and trying to find its solution curves. I have a proposed solution to this which uses the fact that $$ \frac{dy}{dx} = ...
0
votes
1answer
21 views

Finding solutions using graph

Find the number of solutions of the equation $$|\ln|x|\;|=\sin(\pi x)$$ I know how to draw roughly the graph of $|\ln|x|\;|$. However, at $x=1$, both the graphs will pass through $(1,0)$. How do I ...
1
vote
1answer
58 views

Drawing the trajectories for a non-linear system

Find the critical points of the system and draw the trajectories, indicate if they are stable, asy. stable or unstable. $dx/dt=x(1.5-x-0.5y)\\dy/dt=y(2-y-0.75x)$ My critical points are $(0,0) ...
0
votes
2answers
50 views

How to solve this type of problems?

I'm struggling in solving this equation and tried to use the elimination method but did not work with me. Can anyone please show me how it can be solved? $$ -555=0.862X+0.138Y-0.345Z, \\ ...
0
votes
1answer
26 views

Solving a system in 3 variables problem?

I need an answer for this problem, thanks in advance for the help. Find $x$, $y$, and $z$ from the problem below. \begin{eqnarray*} -2x + 1 &=& 5 \\ \\ 2x + 3y - 4z &=& 7 \\ \\ 3x ...
1
vote
1answer
38 views

bifurcation with more than parameter

Problem: Consider the scalar differential equation depending on the parameters $\alpha_1, \alpha_2$ ∈ $\Re$ $x˙ = \alpha_1 + \alpha_2 x − x^2$. Find a change of coordinates $y = \phi(x)$ such that ...
0
votes
0answers
20 views

A function $w(t)$ which satisfy $\int dt w(t)F[x](t)=c$ for all x

Consider a differentiable scalar function in two variables $F(x,t)$ for $x\in X$ and $t\in T$, then it can be viewed as an infinite family of scalar functions $\{F[x](t))\}_{x\in X}$. Are there any ...
1
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0answers
54 views

Decoupling system of two partial differential equations

If I have the following systems of PDE $$ u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0, $$ where ...
1
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1answer
43 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
vote
1answer
41 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: ...
0
votes
1answer
20 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
votes
1answer
45 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
votes
1answer
17 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
votes
1answer
47 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 ...
0
votes
0answers
18 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
votes
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
votes
1answer
39 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 ...
2
votes
2answers
80 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 ...