All Questions
Tagged with systems-of-equations nonlinear-system
439 questions
7
votes
2
answers
732
views
Strange ODE system
As I was solving a physics problem I came across this very strange ODE. My goal is to get the differential equation for the temperature T. The problem is that I can't get rid of the current I in any ...
- 151
0
votes
1
answer
37
views
About the number of solutions of system of non linear equations
I was solving a problem of Lagrange multipliers and faced the following system of non-linear equations:
I managed to find 6 solutions by setting each of the variable to 0 and then playing arround ...
- 143
-2
votes
0
answers
49
views
A system that I didn't solve
We have that $s_1=a+bi$, $s_2=a-bi$, $L>0$ and $$
s_3\in\mathbb{R},\qquad s_4=s_3+\frac{2\pi}{L}k,\qquad s_5=s_3+\frac{2\pi}{L}(k+l),\qquad k,l\in\mathbb{N}.
$$
I want to know what is $L, s_3, a,b$ ...
- 135
2
votes
1
answer
98
views
System of two symmetric quadratic equations
I am trying to find $(x_1,x_2)$ solving the system of equations
$$
p + a_1 x_1 = \dfrac{k_1}{2} a_1 \left[ p + a_1 x_1 + \left(1 + \dfrac{x_1}{s_1} \right) b_2 x_2 \right]^2 \\\\
p + a_2 x_2 = \dfrac{...
- 57
1
vote
0
answers
29
views
Identity matrix in Levenberg–Marquardt algorithm for overdetermined system of equations
I'm trying to learn using LM algorithms for solving non-linear equation system.
The trail step of LM algorithm is:
$$ x_{k+1} = x_{k}-\ (J_{k}^{\ T} J_{k} + \lambda_k I)^{\ -1} \ J_{k}^{\ T}\ F_{k} $...
- 141
2
votes
1
answer
64
views
Is every finite collection of points in $\mathbb{C}^n$ the solution to a compatible algebraic system?
Preliminaries:
Bezout's theorem tells us that if we have a system of $n$ multivariate polynomials over $\mathbb{C}$
$$P_1(x_1 .. x_n) = 0 \\ \vdots \\ P_n(x_1 ... x_n) =0 $$
Of degrees $d_1 ... d_n$ ...
- 17.9k
0
votes
0
answers
37
views
Gauss-Newton method diverge for overdetermined system of equations, would Levenberg–Marquardt work?
Problem set up:
Consider the system of non-linear equation that I have
$F(X) = (F_1(X),F_2(X),.....,F_m(X))^T = 0 $
where $\ X = (x_1,x_2,x_3)\in {\rm I\!R}^3 $ and $ F_i(X) = x_1 + x_2 * a^{x_3} + b ...
- 141
1
vote
1
answer
183
views
Finding a Symbolic or Approximate Symbolic Solution for an Unknown Constant in a Complicated (Potentially Transcendental) System of Equations
The following set of equations describe a physical system, where $A$, $B$, $C$, $x_0$, and $z_0$ are known constants (with $A,B,C > 0$, $z_0 < 0$, and $C \ll 1$ but non-negligible), and $\beta \...
- 79
1
vote
1
answer
63
views
Solve 1 Exponential Equation With 3 Initial Conditions [closed]
I have an exponential equation,
$$
\begin{align}
y = A - Be^{kx}
\end{align}
$$
where $(x,y) = (0,3500)$, $(x,y) = (6600,100)$, and $(x,y) = (6000,1000)$ are the initial conditions (IC).
I tried ...
- 102
2
votes
0
answers
48
views
Simplifying complex system of equations (n equation with 3 unknown) to linear system of equations? [closed]
I am working on simplifying a system of non-linear equations to linear one by substituting my unknown with new ones. I realized from my previous attempts that I should not express new unknown, e.g., $...
- 141
-3
votes
2
answers
90
views
How to solve a simultaneous equation with composite hidden variables $xx, xy, yy$? [closed]
Is there a general method for solving a system of simultaneous equations, where each term is always a composite product of 2 atomic variables, e.g.
$$xx - \frac{3}{7}xy + \frac{1}{7}yy = 0$$
$$xy - ...
- 848
0
votes
0
answers
63
views
Constructing a System of Two Cubic Polynomial Equations with Exactly 9 Real Solutions in Maple
I am trying to construct a system of two cubic polynomial equations in two variables (x and y) with exactly 9 real solutions using Maple. However, I am having trouble finding the appropriate ...
- 23
1
vote
0
answers
67
views
Simplifying equation to a simplistic linear form!
Is there a way to write this equation
$$X + a * b^{\frac{1}{Y}} * Z = c$$
in the linear form
$$a'X + b'Y+ c'Z + d' = 0$$
Where $a,b,c,a',b',c',d'$ are known variables; while $X,Y,Z$ are the unknown ...
- 141
2
votes
1
answer
90
views
Solving the system $\frac{xy}{ay+bx}=c$, $\frac{xz}{az+cx}=b$, $\frac{yz}{bz+cy}=a$ for $x$, $y$, $z$ [closed]
I came across a question regarding sytems of linear equations. I have tried elimination,substituition and simon's factoring trick etc but still not able to extract x,y,z.
$$
\begin{cases}
\dfrac{xy}{...
- 51
-1
votes
1
answer
56
views
Help solving a simple nonlinear system of equations [closed]
I need some help making sure Im solving a nonlinear system of equations correctly.
The system in question is the following:
$
\begin{cases}
x(y+1)(z+k) &= 0\\
\alpha(y+1)(z+k) &= 0\\
\alpha x (...
- 35
3
votes
2
answers
138
views
Solving the system $x^4+y^4+z^4=a$, $xy+xz+yz=b$, $xyz=c$
I am trying to solve the following system of equations:
$$
\begin{cases} x^4+y^4+z^4=a\\[4pt] xy+xz+yz=b\\[4pt] xyz=c\end{cases}
$$
where $a$, $b$ and $c$ are constants and $x$, $y$ and $z$ are the ...
- 31
0
votes
1
answer
44
views
Solving a System of Equations Involving Complex Variables and Their Magnitudes
I’m trying solving a system of equations with two complex variables, x and y . The equations are given by:
$$ x + y = c_1 \\ |x| + |y| = r
$$
where $c_1$ is a given complex number and r is a ...
- 129
0
votes
0
answers
23
views
A function of specific form passing through two given points
Let
$$s(t; a_0)=a_{0}t^{2}\left(\frac{1}{2}-\frac{t}{3T(a_0)}\right)$$
with $T(a_0)=\sqrt{\frac{6d}{a_{0}}}$ (where $d$ is some positive real constant).
Then, let
$$
s^*(t; a_0, t_w) = s\left(\frac{t-...
1
vote
0
answers
27
views
invertibility of overcomplete system of non linear equations
I am currently working on a research problem involving 8 nonlinear equations in 5 variables. While these variables are all real, the equations themselves are complex in general. A colleague has ...
- 129
-2
votes
2
answers
47
views
simultaneous non linear equations [closed]
$x^3y^3(x^3+y^3) = 905$ and $x^4y^4(x+y) = 810$. Find values for x and y.
Divide the first equation by the second equation, and factor to give $(x+y)^2 = \frac{25}{6}$. If I had stopped at this point, ...
- 31
3
votes
0
answers
144
views
Non-autonomous system of two nonlinear ordinary differential equations with conditions
Consider the ODE system:
$$
\frac{df}{dx}= -\sqrt{g},\tag{1}
$$
$$
\frac{dg}{dx}= -\sqrt{x}f,\tag{2}
$$
where $f=f\left(x\right)$ and $g=g\left(x\right)$ are the functions on the interval $x\in\left[0,...
0
votes
0
answers
38
views
Method for solving polynomial system without multilinear form?
I am an engineer who is currently working with some network optimization problem during my post graduate study. During my study time, I see that sometimes I need to look for solution of polynomial ...
4
votes
1
answer
343
views
On the solution of these equations
Do not forget to see the Good News at the end of the problem.
This problem is linked to the previous one, up to a changes of coordinates. However, that question is actually about only the first three ...
- 133
1
vote
3
answers
295
views
How to know if there are real solutions or not?
Consider the following system of equations:
$$\begin{cases}
1 + 6(x+y) + 8xy = 0 \\ y = 6x^2 - 8 xy^2
\end{cases}
$$
If for example from the first one I solve for $x$, and then I substitute into the ...
- 3,521
0
votes
0
answers
16
views
Find unknown vector knowing products of its subsets
I would like to solve the following problem. I have a $N\times 1 $ complex vector, for example with $N=6$:
$$v=(a \; b \; c\; d \; e \;f)^T$$
and its complex conjugate
$$\bar{v}=(\bar{a} \; \bar{b} \; ...
- 363
1
vote
1
answer
115
views
To prove the "symmetry" of the solution to equations
$x$ and $y$ are real numbers (edited: thanks to @mcd's answer, now the "positive" requirement is removed) constrained by $ax+by=c$ where $a,b,c$ are strictly positive number.Given a strictly ...
- 85
-1
votes
2
answers
55
views
Solving inequality of multiple variables
What should be a general approach towards solving these kind of questions algebraically?
I tried to code this up and find a solution but I don't know whether it's correct or not. As I've used a scipy ...
- 39
0
votes
1
answer
91
views
Linear programming with $XX^T=Identity$ constraint
I have the following system of equations
$$X R a - c_1 = 0$$
$$X a - c_2 = 0$$
$$X X^T=Identity$$
where $X\in\mathbb{R}^{3x3}$, $a,c_1,c_2\in\mathbb{R}^{3x1}$, $R\in\mathbb{R}^{3,3}$ is a rotation ...
- 107
0
votes
0
answers
49
views
Modified IVT theorem in $\mathbb{R}^2$
Poincaré–Miranda theorem in $\mathbb{R}^2$: $f,g:[-1,1]\times[-1,1]\to \mathbb{R}$ continuous if $f(-1,y)<0,f(1,y)>0, g(x,-1)<0$ and $g(x,1)>0$ then there exists $(x_0,y_0)$ such that $f(...
1
vote
1
answer
82
views
Linearization of a nonlinear third order ODE and stability [closed]
I would like to know if the following differential equation ($\alpha,\beta,\gamma,d,\Lambda,w$ are constants)
$x'''(t)=\frac{1}{24 (3 \alpha -\beta )}\frac{x(t)^{-3 w-2}}{x'(t)} \left(36 \alpha x(t)^{...
2
votes
1
answer
103
views
Solving a coupled system of ordinary differential equations
I want to find two functions $f_1(t)$ and $f_2(t)$ such that
$$
\log f_1=A_{11} \log\left(-\dot{f_1}+a_1 f_1\right) +A_{12} \log\left(-\dot{f_2}+a_2 f_2\right) \\
\log f_2=A_{21} \log\left(-\dot{f_1}+...
- 1,420
0
votes
1
answer
61
views
A geometric-like distribution: determination of the parameter
Suppose that $M > \mu > m \geq 0$ are all fixed integers and assume that ${\bf p} \in \mathcal{P}(\mathbb N)$ is a probability distribution with support on $\{m,m+1,\ldots,M\}$ given by $$p_n = ...
- 2,882
1
vote
1
answer
78
views
System of equations where x and y are real numbers
Solve in $\mathbb{R}^2$ the system of equations:
\begin{aligned}
3^x - \frac{1}{y^2} &= 25 \\\\
\log_9(x) - \log_2(y) &= 1
\end{aligned}
We can rewrite the second equation as
$\log_3(x) + \...
- 679
1
vote
0
answers
44
views
Is it Possible to Treat a Quadratic System of Equations as Linear to Show that the Equations are not Independent?
I have a system of $N$ equations with $n$ complex variables. Each equation has the form $a x_n^2 + b Re(x_nx_{n-1}) + ...+ c x_{n-1}^2 + d Re(x_{n-1}x_{n-2}) +...+ fx_1^2 = H$, where $H$ is a real ...
- 21
0
votes
0
answers
42
views
How to show that a nonlinear system of equations has a unique solution
Consider the system below of 13 equations with unknowns $\alpha, p_1, p_2, p_3, p_4, q_1, q_2, q_3, q_4$, each belonging to $[0,1]$. Could you help me to show that this system has a unique solution?
...
- 310
3
votes
0
answers
65
views
What are the maximum number of solutions to a system of non-linear inequalities?
What are the maximum number of solutions to a system of non-linear inequalities? In particular:
Let $A, B, C$ be real nonzero numbers. Consider the set $S = \{\frac 1 A, \frac 1 B, \frac 1 C, \frac {...
- 5,275
0
votes
0
answers
25
views
Solving systems of simultaneous equations with permutations as variables: algebra meets combinatorics and order statistics
We may treat a given permutation as a vector variable, so that permutation s$_n$ is a variable that takes values among all possible number sequences of length $n$, which have elements in the rank ...
- 171
0
votes
1
answer
105
views
Exact solution to system of first-order coupled nonlinear ODEs
I have recently been trying to find an exact solution to the following system of first-order ODEs:
$\begin{cases} \frac{dx}{dt}=y(t)*z(t) \\ \frac{dy}{dt}=x(t)*z(t) \\ \frac{dz}{dt}=x(t)*y(t)\end{...
- 436
0
votes
1
answer
38
views
Using a system of non-linear equations to prove an identity
Prove that if $3xy + 2yz + z + 1 = 0$ and $3zx + 2z + x + 1 = 0$, then $3xy + 2x + y + 1 = 0$.
I think the solution will involve combining these two equations in some way. I have attempted many ...
- 1,970
0
votes
1
answer
61
views
Solving a "simple-ish" system of nonlinear equations
I have a somewhat seemingly simple set of nonlinear equations that need to be solved given by
$$
\begin{array}{ccccccc}
x & + & m_1 y & + & m_2 z & = & 0\\
\...
- 364
1
vote
0
answers
28
views
How to find the angles required to construct an N-member long chain to a given point?
I have a chain of line segments (vectors also work) such that every line segment begins at the end of another, save for the first which begins at a given point $P_0$, such as in the picture (1), and ...
- 83
1
vote
2
answers
77
views
How to prove that if $x^2+c\sqrt{b+x^2}=y^2+c\sqrt{b+y^2}$ then $x=y$?
I have a function of $x^2$ as $f(x^2)=x^2+c\sqrt{b+x^2}$ and $f(y^2)=y^2+c\sqrt{b+y^2}$ now suppose $f(x^2)=f(y^2)$, How can I prove that $x=y$ if there is a constraint that both $x$ and $y$ are ...
- 23
0
votes
1
answer
87
views
Solution to the system $a_1\sin\theta+b_1\cos\theta=x+c_1$, $a_2\sin\theta+b_2\cos\theta=x+c_2$ does not satisfy initial equations
I have the following system of equations:
\begin{align*}
a_1\sin(\theta) + b_1\cos(\theta) &= x + c_1 \\
a_2\sin(\theta) + b_2\cos(\theta) &= x + c_2 \\
\end{align*}
And I am trying to solve ...
1
vote
2
answers
81
views
Solving a system of nonlinear ODEs [closed]
I want to solve a system of nonlinear ODEs:
\begin{cases}
x_1' = -\dfrac{x_2}{x_3^{2}} \\
x_2' = \dfrac{x_1}{x_3^{2}}\\
x_3' = 1
\end{cases}
For $x(\frac{1}{\pi}) = (0,-1,\frac{1}{\pi}$)
I definitely ...
- 717
0
votes
0
answers
99
views
System of nonlinear first-order PDEs
The following system of nonlinear first-order PDEs describes the
one-dimensional incompressible flow of an ideal fluid in an open long channel
$$h_t+(hv)_x=0,$$ $$v_t+vv_x+gh_x=0,$$ where $h = h(x, t)$...
1
vote
1
answer
68
views
Solving a System of Equations with $\sqrt3xz + yw$
I'm looking to solve the following system of equations and find the value of $\sqrt3xz + yw$:
$\begin{cases}
3x^2 + y^2 - 3xy = 3 + 2\sqrt 2 \\
y^2 + z^2 - yz = 9 + 6\sqrt 2 \\
z^2 + w^2 + \sqrt 3zw = ...
- 11
1
vote
1
answer
57
views
Zero set of system of two real quadratic forms
Background: Consider the equation $x^T A_1 x = 0$ where $x \in \mathbb{R}^\mu$ and $A_1 \in \mathbb{R}^{\mu \times \mu}$ is a symmetric matrix. Suppose we also demand the normalization $x^T x = 1$. ...
- 720
2
votes
1
answer
155
views
System of Quadratics Forms - when is a solution guaranteed to exist?
Suppose I have a system of $\nu$ quadratic forms,
\begin{align*}
x^T A_1 x &= 0 \\
x^T A_2 x &= 0 \\
&\vdots \\
x^T A_\nu x &= 0,
\end{align*}
where $x \in \mathbb{R}^{\mu}$ and each $...
- 720
1
vote
0
answers
126
views
Preimage of a point under the Hopf map
Considering the Hopf map $S^3 \longrightarrow S^2$ given by $$(x,y,z,w) \mapsto \bigl(x^2+y^2-z^2-w^2,2(xw+yz),2(yw-xz)\bigr),$$
I know that the preimage of point is a great circle in $S^3$. For ...
- 23
1
vote
2
answers
110
views
Why do two equivalent systems of equations produce a different result?
I've been reading a book about material and energy balance when I faced a confusing problem when solving a system of equations. I came to notice some new concepts (that were already there, but didn't ...
- 63