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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0answers
60 views

Proving a equation from a System of 5 equations

This equation emerged while I was solving a geometry problem, which was equivalent to proving $x+n=m$. Solve the system of polynominal equations where $x,y,z,m,n,l$ are positive reals. $$ ...
2
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1answer
46 views

Find solution to matrix equation

Find all solutions to the equation $X^2+ \begin{bmatrix} 1 & -1 \\ 1 & 1 \\ \end{bmatrix}X+\begin{bmatrix} -7 & 1 \\ 0 & 0 \\ ...
0
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0answers
54 views

Can anybody explain me how to solve logical equations with using matrices?

Task:I need to find N, with which number of solutions is 32. \begin{cases} (X_1 \land X_2) \oplus (X_1 \land X_3) \oplus (X_2 \land X_3) =X_1 \land (\lnot( X_2 \land X_3)) \\ (X_2 \land X_3) \oplus ...
3
votes
1answer
15 views

Simple system of two nonhomogeneous ordinary differential equations solved by elimination. (3.1-15)

My differential equations textbook states to use the "elimination method" to crack this for $x$ and $y$. The final answer uses $t$ as the independent variable which both $x$ and $y$ are dependent on. ...
1
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0answers
16 views

Simple system of two nonhomogeneous ordinary differential equations solved by elimination. (Ex 3.1-2)

Update: I was able to solve it on my own after all. This problem is actually an example in my differential equations textbook which I cannot seem to duplicate. My textbook states to use the ...
0
votes
1answer
25 views

problem in solving system of equations in Matlab

please help me to find the problem. I ran the below code syms u v S = solve([2*u + v == 0, u - v == 1], [u, v]) and got the below error ...
0
votes
0answers
21 views

What is the sufficient and necessary conditions to have at least one solution of a system of three linear equations?

I am looking for the sufficient and necessary conditions to have at least one solution of the following system of linear equations \begin{align} -\tau \omega x_1 + x_2 + px_4 + c - y &= 0 \\ x_1 ...
0
votes
0answers
34 views

Controlling a flying vehicle with multiple thrusters

I'm working on a problem involving a vehicle with $n$ rocket engines, as seen here: The task is, given the desired force $\vec F$ and torque $\vec \tau$, calculate the optimal thrust for each ...
1
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2answers
28 views

System of linear recurrences

During some computations I came up with the following system of linear recurrences: $$B_{n+2} = 3B_n + A_n \\ A_n = A_{n-1} + B_{n-1}$$ Here I am trying to find the solution for $B$ (hoping to get ...
0
votes
4answers
66 views

Can we solve an algebraic system where the number of equations is less than the number of unknowns?

Is it necessary to always have the number of equations >= number of unknown variables to solve the problem? Can we have the question where this is not true?
0
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1answer
53 views

Solving system of ordinary linear differential equation

I want to solve the following system of differential equations $$\dot x_0(t)=-ax_0(t)+\mu x_1(t)$$ $$\dot x_n(t)=(\lambda(n+1)+a)x_{n-1}(t)-(a+(\lambda+\mu)n)x_n(t)+\mu(n+1)x_{n+1} \qquad \forall ...
0
votes
0answers
15 views

How to solve for the constants of a non-linear equation?

I don't know the correct method to solve for the constants in equations like these (when I am trying to find the solution to a trial non-homogeneous recurrence): $$a\cdot n^2 + b\cdot n^3 + c\cdot ...
1
vote
1answer
27 views

Do (systems of linear equations with scalars and unknowns from different algebraic structures) occur widely?

Generally in linear algebra one studies systems of linear equations where both coefficients and unknowns belong to the same field. I would not be the first person to notice that a system like ...
0
votes
2answers
45 views

If the system $Ax=0$ is consistent, is $Ax=b$ consistent $\forall b$? [closed]

If the linear system $Ax=0$ ($A_{3\times 3}(\mathbb{R})$) has infinitely many solutions, does it mean that the system $Ax=b$ is consistent $\forall b\in\mathbb{R^3}$. If not, find the set of ...
0
votes
0answers
10 views

How to determen the coefficients of a vector equation

Let $\alpha_i \in \mathbb{R}$ real coefficients and $ x_i,b \in \mathbb{R^n}$ vectors, so that $$b=\sum^{n}_{i=1}\alpha_i \cdot x_i$$ Now here is my question: can we from that form, without making a ...
0
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1answer
15 views

Constant values in solution of differential equation after separation of variables

In a previous question, I was dealing with the following equation $$x_1^2 \frac{d^2 A(x_1)}{dx_1^2}\frac{1}{A(x_1)} + x_1 \frac{d A(x_1)}{dx_1} \frac{1}{A(x_1)} + k^2 x_1^2 = -\frac{d^2 ...
3
votes
2answers
36 views

$(x+y)(x+1)(y+1) = 3$ and $x^3 + y^3 = \frac{45}{8}$

I've came across this problem : If $x$ and $y$ are real numbers such that $(x+y)(x+1)(y+1) = 3$ and $x^3 + y^3 = \frac{45}{8}$, find $xy$. This is what I've tried so far: $$x^3 + y^3 = ...
0
votes
1answer
56 views

Is it possible to solve such system of Boolean equations?

During a discrete mathematics test I got this question. I did not see it in my course and I am baffled, because I don't even know from where to start. All I found on google for such subject were ...
3
votes
0answers
38 views

System of equations for operations

Given a system with multiple equations, where we know the values and the result, but not the operations between the values: \begin{cases} 3 ⊕ 5 ⊙ 2 = 13 \\ 7 ⊕ 2 ⊙ 4 = 10 \\ 4 ⊕ 3 ⊙ 3 = 9 \end{cases} ...
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votes
1answer
26 views

Finding point of intersection between 2 parameterised lines

Given the problem of finding the intersection of 2 parameterised lines L1: $x=2-t ; y=1+t$ and L2: $x=2+t ; y=4+t$. Recovering original eqns $y=3-x$ and $y=2x$ yields the correct answer of ...
3
votes
4answers
58 views

How many integer solutions does the following system have?

$$y_1 + y_2 + y_3 = 20$$ $$1 \le y_1 \le 5$$ $$y_2 \ge 5$$ $$y_3 \ge 5$$ I know how to solve it if it were: $$y_1 + y_2 + y_3 = 20$$ $$y_1 \ge 1$$ $$y_2 \ge 5$$ $$y_3 \ge 5$$ then I would do: ...
5
votes
5answers
1k views

Some equations from Russian maths book. [closed]

Could you please help me with solving these equations. I would like to solve them in the most sneaky way. All of the exercises in this book can be solved in some clever way which I can't often find. ...
0
votes
2answers
44 views

Solve system of equations using matrices

Matrix of Unknowns: $X$ Known values (constants): $A, K, F$ How to solve using any programming language: $$ AX + XA^\top + K\cdot X = F $$ (${}\cdot{}$ is element wise product )
2
votes
0answers
33 views

Why can't we model periodic phenomena using a single autonomous differential equation?

I have the system below. It is used to model the interaction between predator and prey. $$x' = x-xy, y' = -y + xy$$ The solution curves are closed contours about the point $(1,1)$. I determined ...
3
votes
3answers
109 views

Why this system have one solution

Let $b\in (1,2),x\in (0,\frac{\pi}{2})$,if such $$\begin{cases} 2b^2+b-4=2\sqrt{4-b^2}\cos{x}\\ 2b^2-4=2b\cos{(x+\frac{\pi}{18})}-2\sqrt{4-b^2}\cos{\frac{5\pi}{18}} \end{cases}$$ show ...
0
votes
1answer
67 views

Not sure why this is true about matrices, but this isn't if they are commutative [closed]

Let $A$, $B$ and $C$ be three matrices. Although for general matrices $PQ \ne QP$, in this particular case I am told that $AC=CA$. I am also told that $A(B+C) \ne BA + CA$. If I am being told in ...
0
votes
2answers
21 views

Linear system of equations: change in one variable with respect to another

Given a linear system of equations, say with 3 equations in 3 variables $x, y, s$, we can solve for these variables in terms of, say, a constant $c$. Let us assume that $x$ is solved and we get $x = ...
0
votes
0answers
13 views

Input a matrix with integral element in matlab

I am working on stochastic fractional differential equation , like below $$ D^2y+D^{\frac{3}{2}}y+y=1+t ,\space y(0)=0 ,y(1)=2$$I use RBF's (radial basis function) to solve it .so $y \sim \lambda_1 ...
0
votes
2answers
24 views

Equivalence between two systems of vector equations

I need to solve the system $$\nabla^2 \mathbf{u} = \nabla p \\ \nabla \cdot \mathbf{u} = 0$$ in a subdomain of $\mathbb{R}^3$ with mixed boundary conditions, where $\mathbf{u}$ is vector field and ...
1
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0answers
46 views

Integer programming, system of linear inequalities.

I am woring on a problem and I got these inequalities. $t_{01}+t_{11}+t_{21}\ge 4$ $t_{02}+t_{12}+t_{22}\ge 4$ $t_{10}+t_{11}+t_{12}\ge 4$ $t_{10}+t_{01}+t_{22}\ge 4$ $t_{10}+t_{02}+t_{21}\ge 4$ ...
0
votes
0answers
23 views

Can Any System Of Equations Be Solved By Any Variable First?

Suppose we have a system of equations with an arbitrary number of variables, but assume it is solvable for each variable (e.g. the system will have as many equations as variables and will have at ...
0
votes
1answer
24 views

Systems of partial differential equations and the Frobenius theorem

I am trying to solve this. I have completed i and ii. i is a result of the Frobenius theorem, and ii requires dp/dx=dp/dy. I am not sure how to solve the system of partial differential equations ...
0
votes
2answers
26 views

nonlinear system of equations

Could you help me to solve the following system of $3$ equations and $3$ variables? $$\left\{\begin{array} {lllll} 2x+y+z=0\\ x-z-\sin y=0\\ x-y+\sin z=0\\ \end{array}\right. $$
0
votes
1answer
39 views

What is the fundamental matrix solution?

Let $A$ be a 3 by 3 matrix, such that $\dot{x}=Ax$. I am trying to find the fundamental matrix solution. I know that I need to find the eigenvalues and eigenvectors of $A$ which I did but I am not ...
1
vote
1answer
30 views

Let $A$ be an $m \times n$ real matrix and $b \in \Bbb R^m$ with $b \neq 0$ . Then

The set of all real solutions of $Ax=b$ is a vector space. If $u$ and $v$ are two solutions of $Ax=b$, then $\lambda u+ (1-\lambda)v$ is also a solution of $Ax=b$ $\forall \lambda \in \Bbb R$. ...
2
votes
1answer
67 views

Specific system of differential equations

I have the following system of equations: \begin{eqnarray}\frac{dx}{dt} = x(1 - x^2 - y^2) \\ \frac{dy}{dt} = y(4 - x^2 - y^2) \end{eqnarray} I want to prove that if a solutions starts (at time $t = ...
1
vote
4answers
63 views

Simultaneous Equations (Stuck on the algebra)

Question: Solve the following simultaneous equations for real values of x and y $$ \left\{ \begin{array}{l} 9^{2x+y} - 9^x \times 3^y = 6 \\ \log_{x+1}(y+3) + \log_{x+1}(y+x+4) = 3 ...
4
votes
1answer
60 views

How to find out the value of $n$ in the given expression.

How can I find out the given expressions value of n? $$\frac{a+b}{2}=\frac{a^n+b^n}{a^{n-1}+b^{n-1}}$$ I tried multiplying both sides by denominator, but it didn't work. Also, observing tells me one ...
1
vote
3answers
25 views

Find the area of the region determined by the system: \begin{align} y & \ge |x| \\ y & \le -|x+1| +4 \\ \end{align}

Find the area of the region determined by the system: \begin{align} y & \ge |x| \\ y & \le -|x+1| +4 \\ \end{align} My attempt Assuming $x>0$ I have the system ...
0
votes
0answers
35 views

How to solve these equations with 4 equations and 3 variables?

Let $X, Y,$ and $Z$ be the variables of interest. $$X = Y + Z + C_1 \\ Y = C_2X \\ Z = -C_3 \\X = f(C_4)$$ They are presented in an economics book as an exercise, and I can't make sense of it. If $X$ ...
0
votes
1answer
51 views

Linear approximation of a stable manifold

Given $$ \begin{cases} \dot{x} = -x + y^2\\ \dot{y} = x - 2y +y^2 \end{cases} $$ Find a linear approximation of the STABLE manifold for the equilibrium $(1,1)$. My attempt: By using the Principle ...
3
votes
2answers
58 views

Find all $(x,y)$ satisfying $(\sin^2x+\frac{1}{\sin^2 x})^2+(\cos^2x+\frac{1}{\cos^2 x})^2=12+\frac{1}{2}\sin y$

Find all pairs $(x,y)$ of real numbers that satisfy the equation $(\sin^2x+\frac{1}{\sin^2 x})^2+(\cos^2x+\frac{1}{\cos^2 x})^2=12+\frac{1}{2}\sin y$ I supposed $a=\sin^2x$ and $b=\cos^2x$ So the ...
3
votes
2answers
52 views

Find $a,b$ for which $xyz+z=a,\quad xyz^2+z=b,\quad x^2+y^2+z^2=4$ has unique solution

Find all values of $a,b$ for which the system of equations $xyz+z=a,\quad xyz^2+z=b,\quad x^2+y^2+z^2=4$ has only one real solution. $xyz+z=a$ $xyz^2+z=b$ So,$\frac{xy+1}{xyz+1}=\frac{a}{b}$ I can ...
3
votes
1answer
92 views

Periodical solutions of this system of differential equations

We have the system of differential equations: $$x'=(1+m)y+x(1-(x^2+y^2))(4-(x^2+y^2)),$$ $$y'=-x+y(1-(x^2+y^2))(4-(x^2+y^2)),$$ with $m>0$. How do I show that $(0,0)$ is the only (instable) ...
1
vote
1answer
27 views

Find critical points of an ODE system

Find the critical points and the solution of the ODE system. Parameters $k_t, \space k_i $ are both positive constants. $$\frac{d \vec u}{dt} = k_t \begin{bmatrix}-1 & 1 \\1 & -1 ...
2
votes
3answers
65 views

Basic question on the infinitely many solutions of a linear system Ax=b,

I just want to verify the geometry of solutions to $Ax=b$, for the case when we have infinitely many solutions: If say for a $3\times 3$ matrix, after Gaussian Elimination, I have two pivot variables ...
0
votes
2answers
124 views

Solve system of equations $3|x|+2y=1, 2x-|y|=4$ [closed]

help with this system of equations \begin{cases} 3|x|+2y=1 \\ 2x-|y|=4 \\ \end{cases} I have no idea.
0
votes
1answer
23 views

Equation about polynomials

Is it possible to find real numbers $x_0, x_1, A_0$ and $A_1$ such equation : $$ (25a+32b+45c+80d) = A_0(ax_0^3 + bx_0^2 + cx_0 + d) + A_1(ax_1^3 + bx_1^2 + cx_1 + d) $$ is true for every real ...
0
votes
1answer
38 views

Solving a system of quadratic equations which evaluates to a 4th grade equation

I have to solve the following system of equations: $x^2 + 4y + 2 = 22$ $2y^2 + x + 6 = 40$ I tried to solve for one variable and then substitute it into the other equation, but a problem appears: ...
0
votes
1answer
35 views

Solve this system of equations explicitly for $f_d$ and $\alpha$

In this system of equations $\alpha$ and $f_d$ are unknown. $\alpha$ and $C_{13}$ are complex numbers; the other parameters are all real numbers. I want to solve this system explicitly for $f_d$ ...