1
vote
0answers
22 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
0
votes
0answers
22 views

Solving system of equations

I have the following set of equations: $y = f(a,b)$ $a = f(y)$ $\dot{b} = f(b,y,\dot{y})$ which I like to solve for $y$. I was wondering if there is some numerical method which I can apply to ...
0
votes
0answers
45 views

Predator Prey Equation

The Predator-Prey Equation is outlined by the following equation: $$\left\{ \begin{array}{l} \frac{dx}{dt}=\alpha x-\beta xy \\ \frac{dy}{dt}=-\gamma y+\delta xy \end{array} \right.$$ Can someone ...
1
vote
0answers
20 views

Removing parametrization from a system of equations

Consider the following system : $$ \begin{aligned} \frac{d^2t}{d\lambda^2} &= -f\left(t\right)\frac{d t}{d \lambda}\frac{d t}{d \lambda} -A\frac{d g\left(t,x\right)}{d \lambda}\frac{d t}{d ...
0
votes
0answers
17 views

System of ODE's with varying times.

Sorry for the vague question, I wasn't really sure how to phrase this. This isn't for homework, it's a problem I am working on. It's been a long time since I've taken differential equations and I'm ...
0
votes
1answer
61 views

Solving system of two linear odes

I am trying to solve \begin{align} y_1'+B_{12}y_1=\beta_{12}y_2\\ Ay_2'+B_{21}y_2=\beta_{21}y_1, \end{align}with $y_1(0)=y_2(0)=y_0$. I find the eigenvalues to be ...
1
vote
0answers
22 views

Solution of a DAE system of two ODE of second degree

I should solve the following DAE system: $$\ddot{x}(t)=-\alpha y(t)$$ $$\ddot{y}(t)=\beta x(t)$$ with the conditions: $x(t)\ge0$, $y(t)\ge0$ and $x(t)+y(t)=N$ with $N\gt 0$. I'm able to solve the ...
0
votes
2answers
35 views

Solving coupled first-order linear ODEs

Basically this question comes from population modelling. Let y be the population of Lions and let x be the population of deer. By ignoring the effect of deer, we observe that $$dy/dt = k_2 y$$ ...
2
votes
2answers
66 views

How to determine the eigenvectors for this matrix

I have the matrix $$\left( \begin{array}{ccc} -\alpha & \beta \\ \beta/K & -\alpha/K \end{array} \right)$$ for which the eigenvalues are ...
0
votes
1answer
51 views

How to solve these two differential equation?

I try to solve these two difference equation ; $$ \frac{dq}{dz} = -j\left(b_1q - kp\right),\\ \frac{dp}{dz} = -j\left(b_2p - kq\right) $$ where $j$ stands for $\sqrt{-1}$, and $b_1$ ,$b_2$ and k are ...
0
votes
0answers
26 views

Nondimensionalization of Coupled ODE

So what I'm messing around with are these two coupled ODES: $$\frac{dx}{dt}=\gamma x\left(1 - \frac{\alpha x+\beta y}{N}\right)$$ $$\frac{dy}{dt}=\theta y\left(1 - \frac{\alpha x+\beta ...
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 ...
1
vote
0answers
21 views

System of ODEs and DAE system

Let us consider the following system of ODEs: $$ y' = f(y,z),\quad z' = g(y,z),\quad y(0) = y_0,\;z(0)=z_0 $$ and the following one: $$ y' = f(y,z),\quad 0 = g(y,z), \quad y(0) = y_0. $$ $f$ and $g$ ...
0
votes
0answers
53 views

Blowing-up a singular point

I have this system of ODEs: $$x'=-y+ \mu x(x^2+y^2)$$ $$y'=x+ \mu y(x^2+y^2)$$ I already find that in $\mathbb{R}^2$ the only singular point is $(0,0)$. So I have to blow-up the singularity to find ...
0
votes
1answer
30 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 ...
0
votes
2answers
34 views

Solving $\left\{\begin{matrix}u'v''-u''v'=0 \\ R^2u'u''+v'v''=0 \end{matrix}\right.$.

Given that $u,v$ are functions of $t$, $R$ constant, solve $\left\{\begin{matrix}u'v''-u''v'=0 \\ R^2u'u''+v'v''=0 \end{matrix}\right.$. When trying to find geodesic on cylinder, I get this ...
0
votes
0answers
34 views

Solving solely continuous system of ode's with matlab

I'm working with the numerical integration of the system of differential equations, $\dot{x}=f(x)$ with the vectorfield, $f(x)$ being solely continuous. Examples of the systems which I'm working on ...
3
votes
3answers
100 views

Need help with simple system of differential equations

thanks to your help I advanced in computing differential equations, but now I encountered another problem I need help with - this time it is a system of differential equations: $$x_1'=-x_2$$ ...
3
votes
1answer
28 views

How to express $z'(t)$ and $w'(t)$ in terms of $z(t)$ and $w(t)$?

I have these functions: $x' (t) = −5x(t) + 2 y(t)$ $y' (t) = 2x(t) − 2y(t)$ where $x(0)=10$ and $y(0)=0$ I am also given these 2 functions: $z(t) = x(t) + 2y(t)$ $w(t) = −2x(t) + y(t)$ First ...
1
vote
0answers
66 views

Solving a system of linear ODEs

Based on my previous post, I have been stuck on this for a few hours now. I want to solve for $x$ and $y$ from the equation $$\frac{dx}{dt} + \frac{dy}{dt}=a-(b+c+d)y-bx.$$ The original two equations ...
0
votes
0answers
29 views

2 Coupled variable-coefficient linear ODEs

I am trying to solve the following boundary-value problem for functions f(x) and g(x): $$ f'' + p_1\left[ f(1)-f(x) \right] + a(x) g - p_2(1-x) -p_6 = 0,\\ (c_1+c_2p_1)g'' - c_3 g^{(iv)} -a(x) f'' =0. ...
1
vote
2answers
33 views

Finding a Lyapunov function for a given system of equations

I've got the following system of equations: $$ \begin{cases} x_1'=-8x_1^3-x_2 \\x_2'=-4x_2-4x_1^3 \end{cases} $$ I'm trying to check, if the equilibrium point in $(0,0)$ is stable or not. I am ...
2
votes
2answers
39 views

where did I go wrong in solving this sytem of nonlinear first-order ODEs?

To communicate my experience level and intent: I'm an undergraduate, this is not homework, I'm trying to write a physical simulation for fun and xp and am stuck just before (what looks to me like) the ...
0
votes
2answers
29 views

Classify critical point of linear system

For this linear system: $\dfrac{dx}{dt}=x+y-2$ $\dfrac{dy}{dt}=x-y-4$ I've found the critical point to be $(1,1)$ but now I want to classify it. How do I do it?
1
vote
0answers
30 views

System of differential equations with references to each other

For system of differential equation as follows:\begin{align} \frac{\partial}{\partial t} \begin{pmatrix}\rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{00}\end{pmatrix} &= -\tau i ...
2
votes
0answers
43 views

How to solve this system of inhomogeneous differential equations

In some past exam papers for the Maths course that I attend,I found this example and I would really appreciate if someone looked at my solution. It goes like this: Find general solution to $$ y_1' = ...
2
votes
2answers
104 views

How to prove that the level sets of this function are closed curves in a specific region.

I need to study the Hamiltonian differential system $$ \begin{align} \dot{x} &= -2ye^{-x^2}\\ \dot{y} &= 2xe^{-x^2}(1-y^2) \end{align}$$ with Hamiltonian function $$ \begin{align} H ...
1
vote
2answers
46 views

Euler method application: step size

Suppose we have a system of ODE's: $a' = -a - 2b$ and $b' = 2a-b$ with initial conditions $a(0)=1$ and $b(0)=-1$. How can we find the maximum value of the step size such that the norm a solution of ...
1
vote
2answers
32 views

System of differential equation with alternative variables

Let us suppose a system of linear differential equations $ \begin{align} \frac{d}{d t}x_1(t)&= -\lambda x_2 (t) \\ \frac{d}{d t}x_2(t)&= \lambda x_1 (t) \end{align} $ How this system could ...
2
votes
1answer
58 views

Stability of zero

Determine stability of zero in \begin{cases} x'=y \\ y'=-f(x) \end{cases} Here $f: \mathbb{R} \rightarrow \mathbb{R}$ is class $\mathcal{C}^1, f(0)=0$ and $xf(x)>0$ for $x \neq 0$. Could you help ...
0
votes
0answers
25 views

Reflection Symmetry for Non-Linear Differential Equations

We are given the equations: \begin{align} \dot{x}& =\mu \, x +y+y^3 \\ \dot{y}& =2x-2y+xy^2+\gamma \, x^2y \end{align} The question at hand is to determine whether there is some sort of ...
1
vote
2answers
58 views

In search of periodic solutions of a system of ODEs by means of Fourier series

Consider the following non-linear system of ODEs : \begin{cases} x' = y \\ y' = x^2-\lambda x. \end{cases} In search of a solution such that $y(0) = y(2 \pi) = 0$, I am being told to seek $x$ and $y$ ...
2
votes
2answers
76 views

How to solve this system of equations that appears in a ODE exercise?

I am trying to solve this equation, we know $A, B, Q,\phi\in\mathbb{R}$. \begin{eqnarray} T''(x) &=& \phi (T(x)-Q) \\ T(0)&=& A\\ T(b)&=&B \end{eqnarray} So the ...
0
votes
1answer
31 views

How to solve this system of the 1st order equations?

This is a problem from the book: $$x_1' = x_2\\ x_2' = -x_1\\ x_1(0) = 2\\ x_0(0) = 0$$ The problem says transform the system of the 1st order differential equations into a single differential ...
1
vote
1answer
41 views

For this 2 by 2 locally linear system, how to determine that this “indeterminate” critical point is a centre? Boyce, p516, Question 9.3.12

$12.$ (a) Determine all critical points of $\dfrac{dx}{dt}=(1+x)\sin y$ , $\dfrac{dy}{dt}=1−x−\cos y$ . (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of ...
1
vote
0answers
43 views

Strong Lyapunov Function

By showing that $V(x_1,x_2) = (x_1)^2 + (x_2)^2$ is a strong lyapunov function for the system: $x_1’ = -x_2$ $x_2’ = x_1 + (x_2)^3 - x_2$ determine a region of ''attraction'' for the origin. I ...
1
vote
1answer
35 views

Show that the solution of the differential system are periodic.

Let $y,z$ two functions defined on $\mathbb{R}$. Show that the solution of the differential system : $$ y'=z^3 \qquad z'=-y^3 $$ are periodic. My attempt : With some works I can show that ...
0
votes
1answer
31 views

Using Laplace Transform to solve a 3 by 3 system of differential equations

I have been trying to solve this system of equations using Laplace transforms for a while. It is very easy to solve it using eigenvalues and eigenvectors, but when I tried to do it using Laplace I ...
4
votes
2answers
118 views

A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$

Let: $f(x)$ and $g(x)$ be twice differentiable, non-decreasing functions. $f''(x) = g(x)$ and $g''(x) = f(x)$. $f(x) \cdot g(x)$ is a linear function. Then we have to show that $f(x) = g(x) = ...
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 ...
1
vote
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
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 = ...
1
vote
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
vote
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
vote
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 & ...
-1
votes
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 ...
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} = ...
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) ...
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 ...
1
vote
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 ...