1
vote
1answer
31 views

solving system of equations(nonlinear)

I am trying to solve the following system of equations: $$\frac{kq^2}{d}=mg(L-L\cos(t))+\frac{kq^2}{r}$$ $$\sin(t)=\frac{x}{L}$$ $$r^2=(L-L\cos(t))^2+(x+d)^2$$ The parameters are: $k,L,d,q,m,g$ The ...
0
votes
4answers
56 views

solving the system

solve the system : $$ y+|x-2|=3 $$, $$ |x+y|= m $$ graphicly when $m$ equals $6$. I can easily (realtively) skecth the first graph , however, how the bloody hell do you sketch $|x+y|= 6$??
0
votes
1answer
23 views

Cramer's rule and linear dependence/independence test

When you have the system of equations: $$ax + by = e\\cx + dy = f$$ And you do some row operations to eliminate $y$, we get: $$x = \frac{ed-bf}{ad-bc}\tag{1}$$ If we eliminate $x$ we get: $$y = ...
1
vote
1answer
23 views

Solving $x' = Ax$ for real $x$ where $A$ is a matrix with complex eigen values

I have the following linear differential equation system: $$x' = A x$$ where $$ A = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 3 & 1 & -2 \\ 2 & 2 & 1 \end{array} \right) $$ I ...
0
votes
2answers
55 views

Find conditions on a, b, c, and d for which the following system has solutions:

Find conditions on $a$, $b$, $c$, and $d$ for which the following system has solutions: $$2x+4y+z+3w=a $$ $$-3x+y+2z-2w=b $$ $$12x+5y-4z+12w=c $$ $$13x+10y-z+13w=d$$ I got the system down to: ...
0
votes
1answer
32 views

Fastest way to compute minimal polynomial (for solving $x' = A x$, $A$ matrix)

In general, given a $3\times 3$ or $4\times 4$ matrix $A$ which doesn't have a lot of $0$ entries, what is the fastest or less error prone way to compute its minimal polynomial? More generally, I ...
1
vote
1answer
24 views

Solve system to find critical points.

Hi I have to find the stationary points for $$f(x)= x^4+y^4-(x-y)^2.$$ So far i founded the partial derivatives for $x$ and $y$. Next step is to solve this system to get my critical points: $$ ...
0
votes
1answer
45 views

Solving system of 3 equations

How do I solve the following system? $$ \left\{ \begin{array}{} x_o = 4 - x_r \\ x_r = -2 - x_s \\ x_s = 2 - x_r \end{array} \right. $$ All the techniques i've found for solving 3-equation ...
0
votes
1answer
32 views

Find the height of the dam given angles of a triangle

The top of a dam has an angle of elevation of 1.3 radians from a point on a river. Measuring the angle of elevation to the top of the dam from a point 155 feet farther downriver is 0.8 radians; assume ...
0
votes
1answer
70 views

Math Constraint Problems

This is a homework problem of mine. The professor said we can use any resource to help us solve and I cannot get up with anyone from class. Please help. I'm not looking for a direct answer, I ...
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 ...
3
votes
2answers
63 views

Solving $L= \frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ priveded $a+b+c=0$

Let $a,b,c$ be such that $a+b+c=0$ and suppose that $$L= \frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}.$$ Find the value of $L$. I can only see the symmetry of these function ...
0
votes
3answers
51 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
votes
1answer
34 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 ...
1
vote
2answers
76 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 ...
5
votes
1answer
194 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 ...
1
vote
2answers
76 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 ...
3
votes
0answers
67 views

An interesting system of equations

We have the following system with a and b, real numbers: $ax+y + z =4$ $x+2y+3z=6$ $3x-y-2z=b$ Show that $\forall a \in \mathbb{Z} $ there is a $b \in \mathbb{Z}$ such that the system admits a ...
0
votes
3answers
57 views

Solve a system of equations involving two ellipses

Problem #38 asks us to solve the system using either graphing, substitution, or elimination. The only way that I can think of doing this is by graphing. However, is there any easy way to solve this ...
0
votes
1answer
23 views

Stability of a System of ODEs Problem

Using theorem 4.2 and problem 1 determine the stability of the following linear system $z'_1 =-7z_1+10t^2(1+t^2)^{-1}z_2$ $z'_2=-(4+t^{-1})z_1+5z_2$ Theorem 4.2: Suppose a constant matrix $A_0$ has ...
0
votes
2answers
31 views

Are there cases in which the Rouché–Capelli theorem does not hold?

I have written a program which solves an $m \times n$ system of linear equations by using Gaussian elimination with pivoting. After I reduce the augumented matrix $[A~|~B]$ to echelon form using ...
2
votes
2answers
29 views

Help with transforming a second order ode into a system of first order ode's

I have the following equation: \begin{align*} \frac{d^2\theta}{dt}=\alpha(\theta-1)+\beta(\theta-1)^3-\gamma\frac{d\theta}{dt} \tag{1} \end{align*} Where $\alpha, \beta, \gamma \in \mathbb{R}$. ...
1
vote
1answer
58 views

Find an appropriate Lyapunov function for the nonlinear system of differential equations

I have tried numerous functions for $V(x)$ but I am having trouble finding one that tells me about the stability at the equilibrium point. Any ideas of one that may work?
0
votes
2answers
98 views

System of linear equations problem

Consider the system of linear equations: \begin{align} \begin{cases} x+ay=1\\[2ex] bx+5y=2, \end{cases} \end{align} where $a$ and $b$ are parameters. (a) Determine the conditions on $a$ and $b$ to ...
0
votes
0answers
34 views

Simple Question- Is this a Sturm Liouville regular problem?

I have the following differential system: \begin{align} &(1-t^2)x'' -2tx' +\lambda x= 0 \\&x(0)=0 \\&x(l)+x'(l)=0 \end{align} I have to decide if it is an homogeneous regular Sturm ...
0
votes
0answers
24 views

Show that when $\epsilon $ < 0, the basin of attraction of the origin contains the region $z > −1$.

Consider the system $$x' = (\epsilon x+2y)(z+1)$$ $$y' = (-x+\epsilon y)(z+1)$$ $$ z' = -z^3$$ Show that when $\epsilon $ < 0, the basin of attraction of the origin contains the region $z > ...
1
vote
1answer
73 views

Understanding for a system why not asymptotically stable when $\epsilon = 0$ at the origin

Consider the system $$x' = (\epsilon x+2y)(z+1)$$ $$y' = (-x+\epsilon y)(z+1)$$ $$ z' = -z^3$$ Show that the origin is not asymptotically stable when $\epsilon = 0$. I am told if we start with ...
1
vote
1answer
46 views

use the definition of ($\epsilon, \delta$) proof to show asymptotically stable?

Compute the solution $\phi_t \overrightarrow x_0 = e^{At} \overrightarrow x_0$ to the system $x' = -x + 4y$ and $y' = -4x - y$. i found the solution that $$\phi_t (x,y) = e^{-t}\begin{bmatrix}\cos 4t ...
0
votes
1answer
14 views

Time-Scale Decomposition: The slow variable only has a single unstable steady state?

I'm completing a time-scale decomposition of a set of equations. The original set of two equations is $$ S'= 3k_1S^3E_T + (3k_{-1}-3k_1S^3)E_1 $$ $$ E_1'=-k_1S^3E_T+(k_{-1}+k_2+k_1S^3)E_1 $$ I was ...
1
vote
2answers
39 views

Given a solution flow to find periodic solutions

Given the system of differential equations $x' = 2x + y^3$ and $y' = -y$ i found the flow $$\phi_t(x,y) = ((x_0 + 1/5y_0^3)e^{2t} - 1/5 y_0^3e^{-3t}, y_0 e^{-t})$$. I am wondering are there any ...
2
votes
1answer
28 views

Compute the solution $\phi_t \overrightarrow x_0 = e^{At} \overrightarrow x_0$ to the system $x' = -x + 4y$ and $y' = -4x - y$

Compute the solution $\phi_t \overrightarrow x_0 = e^{At} \overrightarrow x_0$ to the system $x' = -x + 4y$ and $y' = -4x - y$ The characteristic equation i found is $\lambda^2 + 2\lambda + 17$...The ...
0
votes
0answers
57 views

Coupled mass spring system with damping and initial values

After researching through the web, I can't figure out how to express into a differential equation a coupled mass spring system with damping and initial values. Two masses and two springs, no external ...
1
vote
4answers
90 views

Reducing the System of linear equations

\begin{align*} x+2y-3z&=4 \\ 3x-y+5z&=2 \\ 4x+y+(k^2-14)z&=k+2 \end{align*} I started doing the matrix of the system: $$ \begin{pmatrix} 1 & 2 & -3 & 4 \\ 3 & -1 & 5 ...
4
votes
1answer
62 views

issues with simple algebraic equations

$ab + a + b = 250$ $bc + b + c = 300$ $ac + a + c = 216$ then find $a + b + c = ?$ MY APPROACH: (i) * c , (ii) * a , (iii) * b then we get $abc + ac + bc = 250c$ $abc + ab + ac = 300a$ ...
0
votes
1answer
80 views

Systems of linear differential equations - eigenvectors

Solve the following system of equations $ \begin{cases} x_1^{'}(t)=x_1(t)+3x_2(t) \\ x_2^{'}(t)=3x_1(t)-2x_2(t)-x_3(t) \\ x_3^{'}=-x_2(t)+x_3(t)\end{cases} $. First, I create the column vectors ...
2
votes
3answers
165 views
3
votes
1answer
70 views

For $x' = x^2$, and $y' = x + y$, Find all equilibrium points and decide whether they are stable, asymptotically stable, or unstable.

For $x' = x^2$, and $y' = x + y$, Find all equilibrium points and decide whether they are stable, asymptotically stable, or unstable. I found that the equilibrium points are (0,0). After linearized ...
3
votes
1answer
45 views

For $x' = y$, and $y' = -x - y$, Find all equilibrium points and decide whether they are stable, asymptotically stable, or unstable.

For $x' = y$, and $y' = -x - y$, Find all equilibrium points and decide whether they are stable, asymptotically stable, or unstable. I found that the equilibrium points are (0,0). Then I try to find ...
2
votes
2answers
50 views

Have I done something wrong in solving the following pair of equations?

Question given: Solve,$3x^2-5y^2-7=0\\3xy-4y^2-2=0$ What I have done so far: $$ ...
2
votes
4answers
92 views

Find the equation of a circle passing three points (conics)

Problem: Determine the equation of the circle that passes through three points, $J(-3, 2)$, $K(4, 1)$, and $L(6, 5)$. I thought of using systems like so: $$\left\{ \begin{array}{rcl} (x+3)^2 + ...
2
votes
4answers
110 views

How to solve the following system of equations.

$2x^2-3xy+2y^2=2\frac{3}{4}\\x^2-4xy+y^2+\frac{1}{2}=0$ I tried all the methods that I know, but I could't isolate $x$ or $y$ to form one equation.
1
vote
0answers
30 views

Let $r_t+ru_x+ur_x=0$ and $u_t+uu_x=0$ for $u(x,0)=f(x)$ and $r(x,0)=g(x)$.

Let $r_t+ru_x+ur_x=0$ and $u_t+uu_x=0$ for $u(x,0)=f(x)$ and $r(x,0)=g(x)$. I know I have to solve for implicit $u$ first and then $r$, but I don't know if I'm using the right method here. What I've ...
0
votes
1answer
46 views

Consider the system $x'= \frac{-x}{2}; y' = 2y + x^2 $ to solve the system and find topologically conjugacy and show topologically conjugate

Consider the system $x'= \frac{-x}{2}; y' = 2y + x^2 $ Show that this system is topologically conjugate to the linear system $\overrightarrow {y'}$ = $DF_{(0,0)}$ $\overrightarrow {y}$ a) Solve both ...
2
votes
1answer
79 views

Find solution for the following system: $x'=y^3-4x$ , $y'=y^3-y-3x$

Find solution for the following system: $x'=y^3-4x$ , $y'=y^3-y-3x$ As I found the three equilibrium points, $(0,0)$ , $(-2,-2)$ , $(2,2)$ . I am wondering how to find the solution for this one ...
0
votes
1answer
50 views

find all the equilibrium points for $x'=y^3-4x$ , $y'=y^3-y-3x$

Consider the system $x'=y^3-4x$ $y'=y^3-y-3x$ What is the systemic way to find all the equilibrium points for this system? My approach is to set both $x'$ and $y'$ to $0$ , solve the two equations ...
0
votes
1answer
50 views

Simplifying a coupled-pendulum equation by assumption

I have been given the following question and I am unsure if I am missing an assumption or if I am misunderstanding something else: Two identitical pendula each of length $\ell$ and with bobs of ...
0
votes
0answers
45 views

How to solve a system of equations with 3 variables?

$$13 = V_{pf} \cdot (0.0518)+V_{bf} \cdot \cos\theta_b$$ $$0=-V_{pf} \cdot (0.9659)+5V_{bf} \cdot \sin\theta_b$$ $$845=V_{pf}^2+5V_{bf}^2$$ Okay, so I'm at the very last part of a word problem and ...
1
vote
2answers
76 views

Why can't you swap rows in the matrix for a system of linear differential equations?

If you are given a Matrix A, and then asked to solve the initial value problem x'=Ax, why can one not swap rows before starting the problem. I tried it with a 3x3 matrix on wolfram alpha and got two ...
1
vote
1answer
59 views

Solution trajectories of a plane autonomous system

I have the plane autonomous system $\dfrac{dx}{dt}=x(1-2x-y)$ $\dfrac{dy}{dt}=y(1-x-2y)$ I need to show that the axes of the phase plane and the line $x=y$ are solution trajectories, but I don't ...