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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2
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2answers
87 views

finding the potential v(x,y)

Consider the system $\dot{x}=3x^2-1-e^{2y}, \dot{y}=-2xe^{2y}$ 1)Show that $\frac {\partial{f}}{\partial{y}}=\frac {\partial{g}}{\partial{x}}$ 2)Find the potential $V(x,y)$ 3)Show trajectories ...
2
votes
2answers
129 views

Unstable fixed point

Consider the system $\dot{x} = x(1-4x^2-y^2)-\frac{1}{2}y(1+x) $ $\dot{y} = y(1-4x^2-y^2)-2x(1+x) $ Show that origin is an unstable fixed point I made $\dot{x} = 0$ and $\dot{y}=0$ and $\dot{x} = ...
1
vote
1answer
96 views

Find the index of the equilibrium points of the system (Question on solution)

I have the following system: $$\dot{x} = 2xy$$ $$\dot{y} = 3x^2-y^2$$ I have the following solution: The system has one equilibrium point at the origin. Let the curve $\Gamma$ surrounding the origin ...
2
votes
1answer
63 views

Simultaneous Equation Problem

$$x + 2y + z = 5(x + y)(y + z)\\ x + y + 2z = 7(y + z)(z + x)\\ 2x + y + z = 6(z + x)(x + y).$$ Find the value of $24^3xyz$. This seems a problem concerning simultaneous equations. I didn't even know ...
0
votes
0answers
24 views

Question on 2 functional equations.

Let $z,x$ be complex numbers. Im looking for analytic functions $f(z)$ such that : $$1) \exp(\ln^{5} (f(x))=\sum_i a_i f(b_ix)$$ $$2)f(x)^5=\sum_j c_j f(d_jx)$$ holds for all $x$ and where both ...
1
vote
0answers
97 views

How to solve analytically or simplify this coupled system of ODEs?

I have a coupled system of ODEs: $$\cases{ i\frac{\text{d}y_1}{\text{d}t}=A f(t)y_2(t)+E_1 y_1(t)\\ i\frac{\text{d}y_2}{\text{d}t}=A f(t)y_1(t)+E_2 y_2(t) }\tag1$$ Here $f(t)$ is a periodic function ...
1
vote
1answer
87 views

Roots of unity and a system of equations by Ramanujan

Is it immediately apparent that the solution to the system of equations, $$\begin{aligned} x_1^2 &= x_2+2\\ x_2^2 &= x_3+2\\ x_3^2 &= x_4+2\\ &\vdots\\ x_n^2 &= x_1+2\\ ...
0
votes
0answers
82 views

Solve system of equations AXB = 0

Is there a common approach to solve a system of linear equations in a form $A^TXB = \bf{0}$? Where $A$ and $B$ are known matrices and $X$ is an unknown matrix. This seems simple enough, there should ...
1
vote
1answer
69 views

Help explain method to solve two equations in two unknowns where one of the variables has a square term

Given: $-\frac{96}{x^2y}+1+y=0$ $-\frac{96}{xy^2}+2+x=0$ Solve for $x$ and $y$ How should I find $x$ and $y$? I thought of using the methods I learned in linear algebra but then I noticed that ...
1
vote
0answers
125 views

gear method for stiff set of differential equation

I have a set of 3 differential equation: $\begin{cases}\dfrac{dy_1}{dt}=s(y_2+y_1-y_1y_2-qy_1^2)\\\dfrac{dy_2}{dt}=\dfrac{1}{s}(-y_2+y_3-y_1y_2)\\\dfrac{dy_3}{dt}=w(y_1-y_3)\end{cases}$ where ...
4
votes
1answer
93 views

Ramanujan's curious cubic identities

Given the cubic, $$z^3-ez^2+fz-1=0$$ Let $z_1,z_2,z_3$ be the three roots. Define, $$x_n = z_1^{1/n}+ z_2^{1/n}+ z_3^{1/n}$$ $$y_n = (z_1z_2)^{1/n}+ (z_1z_3)^{1/n}+(z_2z_3)^{1/n}$$ Ramanujan found ...
1
vote
3answers
113 views

Solving the system $(18xy^2+x^3, 27x^2y+54y^3)=(12, 38)$

While answering this question, I got myself stumped with this crazy system with an evil graph: $$\begin{cases} 18xy^2+x^3=12 \\ 27x^2y+54y^3=38 \end{cases}$$ and I wonder whether there is some slick ...
3
votes
1answer
46 views

Dynamical system equilibrium point increment

I am reading through the dynamical systems theory and there is an example of a Mass-Spring system. The state equations are given by $\displaystyle \frac{d x_1}{dx}(t) = x_2(t)$ $\displaystyle ...
2
votes
3answers
217 views

solve a linear system using gauss-jordan elimination method

We are asked to solve the following linear system $$x_1-3x_2+x_3=1$$ $$2x_1-x_2-2x_3=2$$ $$x_1+2x_2-3x_3=-1$$ by using gauss-jordan elimination method. The augmented matrix of the linear system is ...
3
votes
0answers
39 views

Qualitative dependence of solution to second-order matrix differential equation on eigenvalues

Suppose we have a matrix differential equation in $\vec{x}(t)=\left(\begin{smallmatrix}x_{1}(t) \\ \vdots \\ x_{n}(t)\end{smallmatrix}\right)$, such that: ...
2
votes
1answer
86 views

A Nonlinear System of Trigonometric Equations

folks! I am getting the following system of trigonometric equations which has come as a product of my research. But, I have tried a lot with no success, to solve this system. Can anyone help? Thank ...
2
votes
2answers
332 views

How to fit a sinusoidal function through 2 points with known slopes?

I can define my sinusoidal function as $y(x) = A\sin(B x+c) + D$ or as $y(x) = A \sin(B x) + C \cos(B x) + D$ Now, I have two points with known slopes that I must fit this sine wave to, thus my ...
0
votes
1answer
45 views

Simultaneous equation difficulty minus

I've got a simultaneous equation, and have attempted it. Hope you guys could check my answer out. $\begin{cases}9x-y=-1~......(1)\\-6x+3y=10~......(2)\end{cases}$ $(1)\times3$ : ...
1
vote
2answers
232 views

How to solve complex Simultaneous Linear Equations

Find $a$ and $b$ (1) $a[(1+\sqrt{5})/2] + b[(1-\sqrt{5})/2] = 1$ (2) $a[(1+\sqrt{5})/2]^2 + b[(1-\sqrt{5})/2]^2 = 2$ Is there a smart method to solve this Simultaneous Linear Equations ? Thanks ...
1
vote
0answers
158 views

Solving system if equations containing trigonometric functions with Ti-Nspire

In trying to solve the following system of equation: $20000\times9.81+a\cos b=0$ $a\sin b=6.17\times20000$ Find $a$ and $b$ . It gives me something containing "n2" in bold and I don't know why? ...
5
votes
0answers
128 views

System of 3 equations

I am doing thermal calculation in electronics and when trying to device a general formula for equivalent system resistance to air flow of a part of real system, I ended with this system of three ...
0
votes
2answers
53 views

system of three equations “please help, studying for a midterm”

Thanks in advance if you are reading this... I have a math midterm in several days and i have found an equation i cannot solve. If possible can you please show your steps. I have the misfortune of a ...
3
votes
1answer
92 views

system of 2 linear differential equations with variable coefficients

I have a system of 2 linear diff equations but with a variable coefficients: $$f''(x)+af'(x)+(1+x)g'(x)+bg(x)=0\\g''(x)+ag'(x)+(1-x)f'(x)-bf(x)=0$$ where $a,b$ are some positive constants. I have no ...
1
vote
0answers
107 views

How to solve this system of equation!

How to solve this sytem of equation: $\begin{cases}(\sqrt{a^2+4}+a)(\sqrt{b^2+1}+b)=1\\27a^6+8b=a^3+2\end{cases}$ I tried to analysis equation (1): $\sqrt{a^2+4}+a=\sqrt{b^2+1}-b\Leftrightarrow ...
0
votes
0answers
28 views

Can this system of equations be simplified?

I have a system of two equations over $x$ and $y$: \begin{cases} \lambda \exp(2x\mu_x) \Phi(-y+\mu_y) = \Phi(-y-\mu_y) \\ \lambda \exp(2y\mu_y) \Phi(-x+\mu_x) = \Phi(-x-\mu_x) \end{cases} where the ...
0
votes
0answers
21 views

System of linear equations with modulo elements

Transform the following $3 \times 3$ matrix $A$ with entries in $\mathbb{Z}/5\mathbb{Z}$ into reduced row echelon form (with $[1]$ as Pivot element and $[0]$ above the Pivot element). State out the ...
2
votes
1answer
102 views

unstable zero solution

How can I choose the function $U$ , where $U(0)=0$ and $\dot U$ is positive definite for the system: $$\dot x=y+xy^2$$ $$\dot y=x+yx^2$$ Because I want to show that the zero solution of the given ...
2
votes
0answers
77 views

Find basis of solutions of this linear system.

I am supposed to find basis of the subspace of vector space $ \mathbb{R}^{3} $ of solutions of this linear system of equations: $y = \left\{ \begin{array}{ll} x_{1}+2x_{2}-x_{3}=0 \\ ...
5
votes
2answers
351 views

Solve a linear system with more variables than equations

Suppose that, after a series of elementary row operations the augmented matrix of a linear system with variables $x_1$, $x_2$, $x_3$, $x_4$ is transformed into reduced row echelon form as follows: ...
2
votes
0answers
113 views

Gauss-Jordan is ALWAYS consistent with Cramer`s rule .

When using Gauss - Jordan elimination to solve a system of linear equations, is the solution you get after obtaining a matrix in Reduced Row Echelon Form THE solution or is there any chance that not ...
2
votes
1answer
129 views

Find all solutions for a system of linear equations over a given field

My Problem is: to find all Solutions for the following given System of linear equations over the Field $K = \mathbb{Z}_{/7}$ The System is given with: $$\begin{equation} \begin{split} ...
0
votes
0answers
30 views

Stability of the System of Differential Equations

Given are the following three differential equations: $A'(t) = - 4A(t) +10B(t)C(t)$ $B'(t) = 4A(t) - 10B(t)C(t) - 3B(t)^2$ $C'(t) = 3B(t)^2$ Now I want to show that 1) $A(t) + B(t) + C(t) = 1$ ...
0
votes
1answer
29 views

System of Differential Equations for Particular Initial Conditions

I want to solve the following system of differential equations: x'(t) = x(t) + y(t) y'(t) = y(t) With the intial conditions x(0) = 1 and y(0) = 1 Obviously, y(t) (as well as y'(t)) are equal to ...
1
vote
1answer
40 views

System of ODEs with products

How can we solve the system of differential equations $\dfrac{df(t)}{dt}=-f(t)h(t), \dfrac{dg(t)}{dt}=-g(t)h(t), \dfrac{dh(t)}{dt}=1-(h(t))^2$ The system does not fall to standard ODE methods.
0
votes
4answers
71 views

$x^2+y^2=1, 5x+12y+13=0$ Simultaneous Equations

Can someone solve this for me and show working out? I just can't do it and I don't know why I am getting x and y wrong. It will be very much appreciated. As basic as possible as well please.
0
votes
1answer
37 views

stability in the periodic orbit and in the singular point

Again I have this: $\dot{x}=-y+\lambda x(36-9x^2-y^2)\\\dot{y}=9x+\lambda y(36-9x^2-y^2)\\\dot{z}=-6z-\lambda^2x^2y^2z^3$ I want to analyze the stability in the periodic orbit and in the singular ...
1
vote
1answer
58 views

On a system of PDE

I would like to know what is the set of solutions to the following PDE. I think it consists of just constants, but I need help to prove. Let $f_1(p_1,p_2)$ and $f_2(p_1,p_2)$ be two functions. The ...
3
votes
1answer
45 views

Solution to a trigonometric system

Find the solutions of the system $$\sin a-\frac{\sqrt{3}}{3}\sin b=0$$ $$\frac{\tan 2a-2\tan a}{\tan 2b}\cdot\frac{\tan 2b-2\tan b}{\tan 2a} =1$$ How to work with them ? Thanks
3
votes
3answers
53 views

Gaussian elimination - number of solutions

How do I know how many solutions does a system of linear equations have? I have such system of linear equations: $\begin{cases} (1+b)x+y+z=1\\ x+(1+b)y+z=b\\ x+y+(1+b)z=b^2 \end{cases}$ And I would ...
1
vote
0answers
111 views

stable and unstable manifolds of singular points and periodic orbits

I have this equation $\dot{x}=-y+\lambda x(36-9x^2-y^2)\\\dot{y}=9x+\lambda y(36-9x^2-y^2)\\\dot{z}=-6z-\lambda^2x^2y^2z^3$ and I want to find the stable and unstable manifolds associated to the ...
3
votes
2answers
87 views

Finding the values of $a$ and $b$ [duplicate]

We are given two equations: $$ \left\{ \begin{array}{c} \sqrt{a}+b=11 \\ a+\sqrt{b}=7 \\ \end{array} \right. $$ How to find the value of $a$ and $b$ by solving the equation? All I could do was use ...
1
vote
1answer
68 views

system of differential equations for spread of infection query

I have been given a set of equations describing the spread of infection in a population: $${\frac{dS}{dt}}=-aIS+bI+c-cS$$ $${\frac{dI}{dt}}=aIS-bI-cI$$ where S and I are susceptible and infected ...
0
votes
3answers
113 views

Is $0x_1+0x_2+0x_3=5$ a linear equation?

Is the following equation regarded as a linear equation? $$0x_1+0x_2+0x_3=5$$ The original question is as below: Solve the linear system given by the following augmented matrix: ...
0
votes
2answers
32 views

Percentages and Proportions

Given two integer variables $x$ and $y$. We are given that each integer variable $x$ and $y$ can't be greater than a given integer $z$. The problem: We are given the proportions $a$ and $b$ such that ...
3
votes
2answers
81 views

Constructing a Distributional Solution to the Inhomogeneous C.R. Equations

The question is to find a fundamental solution to the system of equations in $\mathbb{R}^{2}$ \begin{array}{l} u_{x}-v_{y}=f\\ u_{y}+v_{x}=g\end{array} and to express the answer as a $2\times2$ ...
2
votes
3answers
65 views

Satisfying equality between logarithmic expressions

Apologies in advance for any misused terminology, or if this is the wrong place for the question (I think it's okay though). I am given a group of logarithmic expressions such as: $- (a \log(a) + ...
1
vote
2answers
346 views

Solution of a system of linear equations

I tried to solve a system of three linear equations in three unknowns. After a series of elementary row operations the augmented matrix of the system of linear equations becomes ...
1
vote
1answer
66 views

Is it wrong to write a linear system as below?

Suppose we have the following linear system \begin{align} 2x_1+3x_2+4x_3&=1\\ 2x_2-3x_3&=6\\ 0&=0.\\ \end{align} Is it wrong to write the above linear system by including all the zero ...
7
votes
1answer
115 views

How do we solve this system of equations?

$a,b \in \Bbb R$ and $$\frac{a^5b-b^5a}{a-b}=30$$ and $$a^5+b^5 = 33$$ I get that $a^6-b^6=(a-b)63$ But I have no idea how to solve after that. Someone could help me?
1
vote
0answers
36 views

Solving a System of Equations with Cosine

How do I solve a system of equations when there is a cosine. Here is the system: $$ \left\{ \begin{array}{c} a+b=77° \\ \cos(a)=\frac{y}{3.5} \\ \cos(a)=\frac{y+1}{3.5+x} \\ ...