Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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How to solve the following system $\frac{\text{d}x}{\text{d} t} = -Ax + \frac{B}{y} - C$, $ \frac{\text{d}y}{\text{d} t} = -Dx + \frac{E}{y} - F$

Is there a way to analytically solve the following ODE system? $$ \frac{\text{d}x}{\text{d} t} = -Ax + \frac{B}{y} - C\\ \frac{\text{d}y}{\text{d} t} = -Dx + \frac{E}{y} - F $$ Where $A,B,C,D>0$ ...
2
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0answers
18 views

Analytical Solution to an ODE

I was wondering if anyone could help me in finding a general analytical solution to this particular ODE $$f''' + c \bigg[ 3 \text{sech}^{2} \bigg( \frac{\sqrt{c}(x-a)}{2} \bigg) - 1 \bigg] \cdot f' + ...
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1answer
31 views

How do solve this pde problem?

EDIT: I know somehow, we end up with an equation relating the derivative of some coefficients to the rest of the stuff. I'm not sure where this equation, or even the constant that we use to get it, ...
0
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0answers
12 views

Positiveness of energy of differential equation

Let $x(t) : [0,T] \rightarrow \mathbb{R}^n$ be a solution of a differential equation $$ \dot x(t) = f(x(t),t). $$ In addition we have functions $E :\mathbb{R}^n \rightarrow \mathbb{R}$ and ...
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0answers
29 views

characteristic equation and particular solution [on hold]

Could you please tell me how to solve $2x^2 y''+4xy'+2y=10x^2 -6x$ Have to find characteristic equation and particular solution. Kindly help.
2
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0answers
16 views

$\omega$-limit set of a point $x \in X$

I would like to verify whether the following definition of the $\omega$-limit set of a point $x \in X$ is correct: $$\omega(x) = \{ y \in M : \exists \text{ sequence }\{t_j\}, \text{ where } t_j ...
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1answer
32 views

Stuck at successive differentiation problem, suspecting error in question.

This is my very first submission so I wish Hi! to everyone. This is indeed a homework problem that involves using Leibniz's formula for n-th derivative. Although this is uncharted territory to me, I ...
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0answers
8 views

PDEs on higher genus Riemann surfaces, e.g. Klein Curve

I'm trying to solve a PDE on compact Riemann surfaces of genus g > 1. Since these can be obtained as quotients of the upper half plane $\mathbb{H}_2$ by some Fuchsian group $\Gamma$, I suppose it's ...
0
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2answers
26 views

Fundamental set of solutions to a differential equation

Say I have a linear 2nd homogeneous ODE of the form $$y''(x)+p(x)y'(x)+q(x)=0$$ Now I know that the general solution to this will be of the form $$y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)$$ where $\lbrace ...
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1answer
30 views

finding general solutions of second order diffrential equation

find the general solution of $$\frac {d^2y}{dx^2} +9y =18$$ I am not sure how to write it in its complementary form because of the roots one being positive and ...
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0answers
20 views

Solution to second order PDE $u_{xy}-xu_x+u=0$

Given second order partial differential equation $u_{xy}-xu_x+u=0$, where $u=u(x,y)$ find the general solution. I tried to use $u(x,y)=v(ξ(x,y),η(x,y))$ substitution to get ...
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2answers
18 views

Specific solution for ODE

Can somebody explain step-by-step, as I can't understand, how to find the particular solution of the ODE? 1) $y' + y = 1$ 2) $y' + 2y = 2 + 3x$
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2answers
48 views

Solve the first order ordinary differential equation $y'(x)=2x \cos^2 y(x)$

Solve $$y'(x) =2x \cos^2 y(x) .$$ \begin{align} \frac{dy}{dx} &= \ 2x \cos^2 y(x) \\ \frac{dy}{\cos^2 y(x)} &=2x \, dx \\ \tan y(x) &=x^2+k, \qquad\qquad k \in \mathbb{R} \\ y(x) ...
2
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0answers
19 views

Sufficient Boundary Condition to a General PDE on a General Domain

We know that for an ODE of $n^{th}$ order we need $n$ different boundary conditions. In PDEs, for example, for Laplace equation $\nabla^2 U=0$ (which is a second order PDE) we need only one B.C. (e.g ...
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1answer
38 views

Solving a differential equation of the form $y'' = f(x,y)$ [on hold]

The question is $$y''=\frac{c}{x} +\frac{d}{y^2}+r$$ $y=y(x)$ $c,d$ and $r$ are constants.
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1answer
25 views

Determination of ordinary differential equations using Wronskian

I am a bit stuck in this question that I found in my textbook - "Show that the Wronskian of the functions $x $, $x^2 $, and $x^2\log x$ is non zero. Can these be independent solutions of an ordinary ...
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3answers
83 views

How do you integrate $e^{-st}t\cos(t)$?

I'm doing differential equations and specifically studying Laplace Transformations, where of course the Kernel is: $K(s,t) = e^{-st}$ And the Laplace Transformation $\mathcal{L}$ of a function ...
1
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2answers
63 views

Periodic solutions of $x'=x^2-1-\cos t$

Consider $x'=x^2-1-\cos t$. What can be said about the existence of periodic solutions for this equation? I'm not sure if periodic solutions exist, but if they do, they must have period equal to $ ...
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3answers
28 views

Power series solution (Why the constant of the recurrence relation can be chosen arbitrarily?)

Please help me understand this: Solve $y''-xy=0$ First, since there are no singular points, it can be guaranteed that we can always find two power series independent solution, centered at $0$, and ...
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1answer
9 views

Decoupling coupled differential equations with time dependent coefficients

Consider the following system of coupled differential equation. $$\left[ \begin{array}{c} \frac{dc_1}{dt} \\ \frac{dc_2}{dt} \end{array} \right] = \begin{bmatrix} -B & -V(t) \\ -V(t) & B ...
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0answers
22 views

how do I resolve equations that are both dependant on each other

I'm working on a project concerning the ideal power equation of aerodynamic bodies seen here: $$P = \frac{1}{2}C A D v^3 + \frac{W^2}{Db^2v}$$ where $P$ = power, $C$ = coefficient of drag, $A$ = ...
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0answers
15 views

O.D.E. in Homogeneity Lemma

Let $\psi: \mathbb{R}^{n} \to \mathbb{R}$ smooth such that $\psi(x) > 0$ for $x \in B(0,1)$ and $\psi(x) = 0$ for $x \notin B(0,1)$. Let $c \in S^{n-1}$ fix and arbitrary and consider the O.D.E. ...
0
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1answer
21 views

Show that $\frac{dx}{dt}=\frac{1}{14}(15-x)$ given $x=15-12e^{\frac{-t}{14}}$

A biologist is researching the growth of a certain species of hamster. She proposes that the length, $x$cm, of a hamster $t$ days after its birth is given by $x=15-12e^{\frac{-t}{14}}$ Show that ...
1
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1answer
24 views

General solution to differential equation, given a polynomial general solution

I am solving one DE and I have to consider the following: $$(y+ax)^n(y+bx)$$ to come up with a general solution to the following differential equation: $$\frac{dy}{dx} = \frac{10x-4y}{3x-y}$$ I ...
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0answers
32 views

Convergence of $\dot{x}(t) = -\alpha(t)x(t) + b\mathrm{e}^{-\lambda t}$

Let $x(t)\ge 0$ obey the following differential equation: $$ \dot{x}(t) = -\alpha(t)x(t) + b\mathrm{e}^{-\lambda t}, $$ where $b>0$, $\lambda>0$, $\alpha(t)\in\mathbb{R}$ is both lower- and ...
1
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0answers
17 views

How to determine when the Green's function do not exist?

I've been solving some problem which asks us to find the Green's functions for some problems when it exists. Now, there's a theorem which allows us to guarantee that it exists. The theorem is as ...
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0answers
18 views

Finding a Lyapunov function for $u''+u'+\sin u = 0$

Now I need to solve $$\frac{\partial V}{\partial x} = (1 + \frac{\sin x}{y})\frac{\partial V}{\partial y}$$
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1answer
19 views

Is it possible that a PDE solved by two different analytical methods with same Initial and boundary values give different results?

I have developed two models of same scenario. Both models involve a PDE which is solved with same Initial and Boundary conditions. In one model it is solved with Laplace transform and in other with ...
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0answers
18 views

reformulating a BVP into a system of first order ODEs

I need to convert the following bounded value problem $y'''+3y''+2y'^2-5y^2=1$ with conditions $ y(1)=1, y'(0)=1 $ and $y''(1)=0$ Into a system of first order Initial value problems in order to ...
1
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1answer
25 views

Largest interval on which the solution exist of ODE $y'=2(1+y)\sqrt{y}$

If $y$ is the solution of ODE $$y'=2(1+y)\sqrt{y}$$ satisfying $$y(0)=0,y(\pi/2)=1,$$ then the largest interval(to the right of origin) on which the solution exists is $1.[0,3\pi/4)$ $2.[0,\pi)$ ...
0
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1answer
16 views

Proving the interval in which a solution is valid

Question: Verify that both $y_1(x) = 1-x$ and $y_2(x)= \frac{-x^2}{4}$ are solutions of the initial value problem $$\frac{dy}{dx}=\frac{-x+(x^2+4y)^\frac{1}{2}}{2}, \ \ \ y(2)=-1$$ and determine ...
0
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1answer
5 views

How is Dulac's Multiplier selected?

I'm aware that Dulac's (Negative) Criterion states that a system of differential equations of the form $x' = f(x,y), \; y' = g(x,y)$ has no periodic orbits in the plane if we can find some function ...
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0answers
17 views

phase diagram of a 2x2 system of O.D.E.

$$\begin{eqnarray} \dot x&=&a\,x+b\,y,\\ \dot y&=&c\,x+d\,y \end{eqnarray}$$ For repeated eigenvalue case, if b−c is positive, then motion is clockwise; if negative, anticlockwise. ...
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0answers
20 views

Finding the General Solution for a System of Differential Equations with Complex Eigenvalues

I think I might just be having trouble with formatting my answer, because I'm fairly sure my work is right up until this point. The question asks to find the general solution to $$X'= ...
0
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1answer
21 views

two dimensional heat equation

Please I really need some help for this exercise, I can't solve it for any ways... I need to prove the maximum principle for the two dimensional heat equation with zero boundary data. Really I need ...
0
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2answers
11 views

Implementing Adams-Bashforth of order 2 (AB2) algoirthm

Assuming we are given the initial condition for an ODE such that: $$ \begin{cases} x' = f(x,t) \\ x(t_0) = x_0 \end{cases} $$ We are going to solve it numerically using AB2. We know that the ...
0
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0answers
15 views

Euler explicit and semi-implicit

I am given a simple dynamic system with an initial condition: $a(t) = 0.9 - 0.1v(t)$ $v(0) = x(0) = 0$ I want to calculate $x(1)$ with a time step of $\Delta t = 1$ using Euler explicit and semi ...
0
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0answers
25 views

Curve fitting on non-linear ODE data

Background The graph below was generated by a set non-linear ODEs. For those of you who might want to know: It shows the maximum distance achieved by a cylinder when fired at a specified ...
0
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1answer
21 views

solution of second order ODE with non-constant coefficients

What is the general solution of $\frac{d^2y}{dx^2}+P[Q-R\cosh(Sx)]y=0$ where $P,Q,R,S$ are real and positive? I tried transforms but cannot get a solution.
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15 views

Differential Equations - General solution to mathematical model

Mathematical model: $dP/dt = \sigma − \delta$ $\sigma = \alpha*A$. A is the amount of apples $\alpha$ is the amount of pesticides per apple $\delta$ is the rate at which the pesticides decay Can ...
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5answers
143 views

Can we abuse traffic patterns to get home earlier?

I had a heated discussion with my co-worker today, and was wondering if someone here could shed some light on this situation. The post is a bit lengthy, but I wanted to put all my intuition down in ...
1
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0answers
19 views

Why settle for Lagrange Interpolation when doing linear multistep ODE integration?

Say that we have some initial value problem: $y'(t) = f(t,y(t)) ; y(0) = y_0$ with $y_0$ and $f(t,y(t))$ known. If we use Euler's method to numerically approximate the first k points, then we have ...
2
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1answer
16 views

Question about finding eigenvectors for differential equations?

I have a non linear system to analyse and sketch the phase portrait of. At one of the equilibria the Jacobian of the linearised system is given by $$\textbf {J}= \begin{pmatrix} 2 & 7\\ 7/2 ...
0
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1answer
19 views

uniqueness of solutions in the future

I'm asked to proof this: If we have $$f:\mathbb R^d\rightarrow\mathbb R^d, f \in C^1$$ verifying:$$\ f(0)=0 \ $$ $$\ \langle p,f(p)\rangle \leq0 ,\ \forall p \in \mathbb R^d$$ then the initial value ...
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0answers
18 views

A specifc solution to Cauchy-Euler differential equation

Find $\alpha$ such as $y=x^\alpha$ is a solution to the differential equation $$x^2\frac{d^2y}{dx^2}+x(1-x)\frac{dy}{dx}-(x+1)y=0$$ (Oxford (2002), modified) We can derivate and obtaint ...
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1answer
26 views

Solve the DE: $\dfrac{ dy}{dx} =\dfrac{y(x+y)}{x(y-x)}$ [on hold]

I worked it out and my answer came to be $y^3=x^3+3K$. Is my answer correct since I don't have any answer to this question. Thanks
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0answers
16 views

analyze stability of a system of first order differential equations of different types [on hold]

I am working in a mathematical model and I need to analyze the stability of the system of differential equations that define the model, but I don't know how and I am tired of read things that doesn't ...
1
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3answers
55 views

2nd degree differential equation

Can someone please tell me how to solve this differential equation? $${d^2y\over dx^2} +y=\tan(x)$$ I am a beginner in ODE and have absolutely no idea how to proceed. Can you also site a reference ...
1
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1answer
26 views

How to find the straight line paths of saddle points for a nonlinear Hamiltonian system?

I have the system $$\dot{x}=y+2xy\\\dot{y}=-x+x^2-y^2$$ Which is Hamiltonian with $$H(x,y)=\frac{1}{2}x^2-\frac{1}{3}x^3+xy^2+\frac{1}{2}y^2$$ Now I want to plot the phase portrait for the system so ...
0
votes
0answers
21 views

Solving traveling wave usin the shooting method

The spatially-dependent Hodgkin-Huxley equation for a cylindrical dendrite or unmyelinated axon: where $\frac{a}{2\rho}\frac{\partial^2V}{\partial x^2}$ is a diffusion term $a$ is the fiber radius, ...