1
vote
2answers
22 views

Show that there are constants $K$ and $\alpha$ such that $|(e^{At})_{ij}|\leq e^{-\alpha t}K$.

I want to prove that if all eigenvalues of $\textbf{A}$ in the sytem $\dot{\textbf{x}}=\textbf{Ax}$ have negative real parts then there exist constants $K$ and $\alpha$ such that ...
2
votes
0answers
47 views

Combining two differential equations

I have two differential equations that are connected by an equation, $L_1\frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1=\frac{dV}{dt}$ $L_2\frac{d^2I_2}{dt^2} + \frac{1}{C_2}I_2=\frac{dV}{dt}$ $I_1+I_2=I$ ...
2
votes
0answers
16 views

Existenence of the solution for a PDE-ODE system.

I have the PDE-ODE system below: $\frac{\partial c}{\partial t}= D \Delta c - \eta \nabla.(c\nabla v)+g(c,v)$ $\frac{dv}{dt}=-\alpha cv+\xi(c,v)$ with initial conditions and Neumann boundary ...
0
votes
2answers
34 views

critical points, differential equation

I have two differential equations and my assignment is to prove that this system have a unique stationary point. $$\begin{align} \frac{dx}{dt}&=a-(b+1)x+x^2 y\\ \frac{dy}{dt}&=bx-x^2y\\ ...
0
votes
1answer
64 views

How to solve this system of 3 ODE?

I would like to know how to solve this system of differential equation. It consist of 3 ODEs, describing the behavior of an Induction Machine supplied with DC voltage. I a interested to derive the ...
0
votes
2answers
21 views

Finding solutions for system of ODE

How does one find solutions for the system of differential equations of the form $$2x'-5y'=4y-x \\ 3x'-4y'=2x-y$$ ? All I can think of, is finding $x'+y' = 3x-5y$ and then substituting $x'$ or $y'$ ...
0
votes
2answers
26 views

Find solutions for an differential equation system

I have a differential equation system $x_1'(t) = -x_2(t)$ $x_2'(t) = -x_1(t)$ I see that I can write $\dot{x} = Ax$ where $A = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$ The complete ...
0
votes
0answers
32 views

Global existence of ode system without solving it explicity.asdf

Here is the ode system that I am looking at $x' = -y-z$ $y' = x + ay$ $z' = b + z(x-c)$ where a,b,c are positive constants. By the local existence theorem, I know that there is a local solution, ...
2
votes
1answer
70 views

Stability analysis for a system of two differential equations

I have this system of differential equations: \begin{equation} \frac{dx}{dt}=\alpha x-\beta xy\\ \frac{dy}{dt}=\beta xy-\gamma y \end{equation} I need to find the critical points and then do a ...
2
votes
2answers
41 views

How to use differential equations to write $x(t)$ in terms of $y$ and $y_0$?

The equations are: $$ \left\{\begin{array}{rcrcl} x' & = & \mbox{}-a\,x & + & b\,xy \\ y' & = & c\,y & - & d\,xy \end{array}\right. $$ They want me to write an ...
0
votes
0answers
18 views

Solving an equation with boundary conditions to find coefficients

I want to find the unknown constants in the function $f(x,y)=A(e^{-i.k_{x}x}+C_{1}x+C_{2})(e^{-i.k_{y}y}+C_{3}y+C_{4})$, using the following known boundary conditions and auxiliary equation ...
3
votes
2answers
34 views

Finding equilibrium points of differential equation

Given the system $$ x'=xe^{y-3}$$ $$\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space y'=2\sin(x)+3-y$$ Find the equilibrium points and decide ...
1
vote
1answer
30 views

How can I solve this set of linear coupled system?

Consider the matrix $A=\begin{bmatrix} -2k & k \\ k & -2k \end{bmatrix}$ .I have to solve this linear coupled system : $X'' = A.X$, where $X= \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix}$ ...
1
vote
0answers
62 views

Find the fundamental matrix of a system of ODEs?

To linearize a system, in one of the steps I am required to find the fundamental matrix $\Phi$(t) of a system such that $\Phi$(0)=I. The example system my professor used: $\dot{x} = x - y - x^3 - ...
2
votes
0answers
28 views

Solving a system of first order differential equations

So, I have (another) problem with differential equations (from an optimal control problem). I am trying to solve the following system of DEs (is this even a system?): $$ \lambda'(t) = r \lambda(t) + ...
1
vote
3answers
37 views

Techniques for solving coupled differential equations

I am trying to solve a system of coupled differential equations to plot streamlines using Matlab. The equations are these: $$\frac{\mathrm{d}x}{\mathrm{d}t} = -3x - 5y$$ ...
2
votes
2answers
54 views

RL circuit as a system of first-order ODEs

The system is as follows:\begin{align}i_1&=i_2+i_3,\\50\sin t&=6i_1+i_2'+5i_2,\\50\sin t&=6i_1+i_3',\end{align} I have to find $i_2,i_3$. This is my first circuit I'm trying to solve, but ...
0
votes
2answers
27 views

Initial value problem with a delta term

Im having trouble solving this initial value problem. I know how to solve it without the delta-term (C1*e^(lambda*t)*S1 + C2*e^(lambda*t)*S2), but how do i solve it with a delta term?
1
vote
1answer
42 views

Solution of $d^2u/dx^2 + u/A = 0 \ (\text{or } \ C),$ with conditions

Does the following ODE: $$d^2u/dx^2 + u/A = 0 \quad (\text{or } \ C),$$ have a solution with the conditions: $$ \left.\frac{d^2u}{dx^2}\right|_{x=0} = 0, $$ $$u(x=0) = B$$ and $$ ...
0
votes
2answers
62 views

How to find the intersection points of lines that are normal to two curves?

Let I have two curves, \begin{gather} f(x)=\frac{x^3}{4}+1 \\ g(x)=\frac{(x-\tfrac{1}{2})^3}{7}+\tfrac{1}{2} \end{gather} There are zero or more lines that are normal to both curves. In other words, ...
1
vote
0answers
62 views

Long-time asymptotic behaviour of a system of two ODEs

We have the following nonlinear ODE: $$ f' = af-bg -(f+g)^k \bigl(f'(0) +g'(0)\bigr) + f'(0), $$ $$ \bigl(G-T(x)\bigr) g' = -af+bg - g'(0), $$ where $a,b,k,G$ are constants, $f'(0)$ and $g'(0)$ are ...
2
votes
0answers
31 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} = ...
2
votes
1answer
85 views

Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
0
votes
1answer
77 views

Solving for Center Manifold with Parameter

I have a system of ODEs given by $$\frac{dX}{d\tau}=\beta X\left(1 - \frac{X+Y}{N}\right)$$ $$\frac{dY}{d\tau}= Y\left(1 - \frac{X+Y}{N}\right)$$ where $\beta $ is a parameter. How should I ...
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
49 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
22 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 ...
5
votes
2answers
78 views

System of non-linear ODE's

do you have any suggestions to solve analytically the Non-linear ODE system $\dot x=18 x^2 y-3p x^2+6p xy$ $\dot y=18 x^2 y-6p xy $ where $p$ is a real constant. Thank you very much cheers
0
votes
0answers
18 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
24 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
54 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
69 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
34 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
22 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
56 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
36 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
36 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
38 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
101 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
30 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
35 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
38 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
43 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
35 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
51 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' = ...