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

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2
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1answer
78 views

Three-Variable Differential Equation Stability

Discuss the stability of the equilibrium points $(1,0,0)$ and $(1,1,0)$ for the system: \begin{align} x' &= y - y^2\\ y' &= z\\ z' &= x - \cos{z} \end{align} I have found the ...
1
vote
1answer
61 views

How to describe behavior of population system, given by system of ODEs?

So I have a system of equations:$$x'(t)=x(t)(4-2x(t)-y(t))\\y'(t)=y(t)(3-x(t)-y(t)) $$ What I understand so far is: if we have x being the population of prey and y is the population of predators. x ...
3
votes
0answers
79 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
4
votes
0answers
113 views

What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
0
votes
1answer
97 views

Wave Equation Partial Differential EEquation

Basically I got a simple wave equation with an extra twist. The PDE is $\frac {\partial^2 y}{\partial t^2} = c^2\frac {\partial^2 y}{\partial x^2} + L $ with homogeneous boundary condition As usual, ...
1
vote
1answer
70 views

An application of Implicit Function Theorem in differential equations?

Let $f$ be a continuous function from $\Bbb R^3 \to \Bbb R$. By a solution of the differential equation $$f(x,y,\dot{y}) = 0$$ We mean a function $y\colon U \subset \Bbb R \to \Bbb R$ where $u$ is an ...
2
votes
2answers
252 views

Analytic solution to Poisson equation

I need to find the analytic solution to this equation, in order to compare it with solution I get from using a numerical solution. However, I have not been able to find the solution. I think I can't ...
0
votes
1answer
34 views

Central Difference Method

Solve the following using the central difference method: $y(x)= y'+ y + 2x$ where $0 < x < 4$ with $n=4$ subintervals (thus $h=1$). Given that $y(0)=0$ and $y(3)=1$, find $y(1)$. Really ...
1
vote
2answers
68 views

Time dependence of velocity from position dependece of velocity

I know dependence of velocity on position $v(x)$ and I wan't to know dependence of velocity on time $v(t)$ I was thinking that using some chain rules or derivative of inverse it would be possible to ...
0
votes
1answer
25 views

Solve the Initial Value Differential Equations

I split the equation and got y+1 dy = xysinxdx, then I divided the right side by y to get 1 + (1/y) = xsinx dx. I took the integrals of both sides and got y + lny = -xcosx + sinx + c. I don't ...
1
vote
1answer
35 views

quick question with 2nd order linear differential equations

I am solving $y''+4y'+5y=2e^{-2x}cos(x)$ I am working on determining $A$ and $B$ in the particular function. I have the following 2 equations: for the sine part : $-2A+3Ax-3B+Bx=0$ for the ...
2
votes
2answers
59 views

Fourier series of complex diff eq

Can I just use Euler's identity to construct the Fourier Series since it is complex? I was personally thinking I could, but I wanted to be doubly sure.
2
votes
0answers
19 views

For what types of differential equations is the Laplace transform most effective?

I'm reviewing for a final exam and want to make sure I know what tools to use for what situations, and was just wondering if there were situations where the Laplace transform is unusable or less ...
0
votes
1answer
55 views

Need help for this case:

I am learning the artificial potential field method for path planning of mobile robot; artificial potential field method has two components: the first one is attractive force and second one is ...
1
vote
1answer
42 views

Topological structure/graph from a paper

This question is based off a paper titled "On designing heteroclinic networks from graphs." I'm having a difficult time visualizing something "drawn in 4-dimensions" projected down to a 2-dimensional ...
1
vote
1answer
41 views

Help solving differential equations

I would like to know how to classify the following equations: $y''+ 4y'+5y=2e^{-2x}cos(x)$. Is it a second order linear equation?
1
vote
1answer
57 views

Can the following nonlinear first order ODE be solved?

I have tried solving this equation from several manners but no luck. Can it be solved? $$\frac{d f}{d t} = A f^2 +g(t)$$ The solution for the homogeneous is (I think; somebody should confirm) ...
1
vote
0answers
70 views

Lotka-Volterra Problem From Arnold's Ordinary Differential Equations

Problem 1 of section 2.7 of Arnold's Ordinary Differential Equations book asks to prove that the period of the oscillations in the Lotka-Volterra model tends to infinity as the initial condition ...
4
votes
1answer
66 views

Did I do something wrong solving this PDE in MATLAB?

I have the following PDE problem on a practice exam: I have completed the problem using MATLAB to the best of my ability. Here is the code I used ...
5
votes
1answer
160 views

Why does $\frac{1}{r}\frac{dr}{d\theta} = \cot \psi$?

In the discussion of linear fractional equations in Birkhoff and Rota's Ordinary Differential Equations, the authors assert that if we convert a DE of the form $y' = F\left(\frac{y}{x}\right)$ to ...
-1
votes
1answer
43 views

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $-\Delta u=1$ in $\mathbb{R}^n$?

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $\Delta u=-1$ in $\mathbb{R}^n$? I think not, but how can we prove it? Let's assume that such a solution exists. Let $R>0$ and $B_R:=B_R(0)$ ...
0
votes
2answers
91 views

Ordinary Differential Equations self-study reference request

I know there are a lot of reference requests for differential equations textbooks but none seem to be what I need. I'm looking for a book I can use for self study that isn't overly complicated and ...
0
votes
1answer
33 views

solving a partial differential equation

How can I solve the following equation? $$-f_{x}+yf_{xy}+xf_{yy} = c^{'}(x)(-f+yf_{y})$$ where $f=f(x,y)$ is a real function of two variables $x,y$ and $c=c(x)$ is a real function of $x$. I guess ...
15
votes
3answers
512 views

Does Tom catch Jerry?

Tom has Jerry backed against a wall. Tom is distance 1 away (perpendicularly). At time t=0, Jerry runs along the wall. Tom runs directly towards Jerry. Tom always runs directly towards Jerry. Tom and ...
0
votes
1answer
44 views

How to obtain an exact solution to nonlinear second order ODE

I need help in analytically solving this nonlinear second order ODE, $A y(x) + y'(x) \Bigg( B + \frac{C y'(x)}{D y'(x) - y''(x)} \Bigg) = 0$. Any help is appreciated.
0
votes
2answers
96 views

Techniques to solve nonlinear first-order ODEs

I am trying to solve the following nonlinear ODE: $$\frac{dy}{dx} = \frac{1}{x(ayx-b)},$$ where $a, b$ are constants and $a>0$. Moreover, you may assume that $b \neq 0$ if necessary. This ...
1
vote
1answer
34 views

Differentiation under the integral

Now I have this expression. $\psi(\theta)=\text{log}\int_{-\infty}^{\infty}\exp{\{\Delta\theta-f(\nu)\Delta^2\}}h(\Delta)d\Delta$. The expression of $h(.)$ is not given. So $h(\Delta)$ is some ...
2
votes
0answers
26 views

Origin/justification of the condition in variation of parameters?

The method of variation of parameters (on e.g. $y"+py'+qy=g$ that yields $y=A(x)y_1 +B(x)y_2$) requires one to use, in addition to the constraint provided by the actual differential equation, one has ...
1
vote
1answer
81 views

Calculating $a_n$ in $\sum_{n=1}^\infty a_n \sin(\frac{n \pi}{2})=T_0$

I'm looking to solve the following when $T_0$ is a constant: $$\sum_{n=1}^\infty a_n \sin\left(\frac{n \pi}{2}\right)=T_0$$ If it matters this was reached from the following: ...
3
votes
3answers
42 views

Viable method to solving this first order system of linear DE?

Given the following system of differential equations \begin{align} \frac{dy}{dt} &= x \\ \\ \frac{dx}{dt} &= y \end{align} is the following operation allowed? \begin{align} ...
4
votes
2answers
103 views

What are some tips/techniques that might help me solve this (brutal) differential equation?

I've been working on a certain physics problem involving differential equation for two years. I've made some progress on it recently, but I've come across another roadblock, namely an integral that I ...
0
votes
2answers
64 views

Solve differential equation with matrix method

I have the following IVP: $$\ddot{x} + 2\dot{x} - 8x = 4$$ subject to the initial values $$x(0) = 0 \\ \dot{x}(0) = 0$$ I am asked to solve it using matrix method (I don't know if it is the correct ...
0
votes
2answers
28 views

Orbits existing for all time

For part $c)$ I understand why the above argument implies that no solution can ever tend to infinity. However I don't understand why this implies that solutions exist for all time. Why if a ...
0
votes
1answer
67 views

First order differential equation (with a logistic function)

I came across this first order differential equation $$ f'(x) = \left( \frac{1}{x} + \frac{g'(x)}{g(x)} \right) f(x) - c \frac{g'(x)}{g(x)} \textrm{,}$$ where $g(x)$ is this logistic type function $$ ...
1
vote
1answer
238 views

Chemical kinetics using Laplace transformation

I have a simple chemical reaction $A\leftrightarrow B$ with forward rate $k_1$ and backward rate $k_2$. I can now write the differential equation of this system as following. $ \frac{dA}{dt} = -k_1A ...
0
votes
1answer
41 views

Difference Between Lyapunov and Strong Lyapunov Function.

Good Day everyone. I was assigned to show that given an autonomous system of Differential Equations and a function $V$, I need to show that $V$ is Lyapunov function. To show that $V$ is Lyapunov. I ...
1
vote
2answers
39 views

Differential equation - help

How should I tackle this differential equation $\frac{d \ln{y(t)}}{d \ln{t}} = \alpha (1 - \frac{p(t)}{y(t)})$ in the unknown function $y(t)$ ? Separation of variables maybe? Thanks to anyone who ...
1
vote
0answers
33 views

Simple RK4 measure of a force in 2nd order ODE

Consider that I am solving a second order ODE using RK2/RK4. The ODE represents simple equations of motion: Equations of motion I am trying to solve: \begin{align} \frac{dx}{dt} &= v \\[.3em] ...
5
votes
2answers
105 views

Homogeneous differential equation - cannot manipulate equation

this was a problem from a textbook: If $x>0$, $y>0$, find the general solution to the differential equation, $$ x \frac{dy}{dx} = y + \frac{x}{\ln y - \ln x }$$ giving your ...
1
vote
1answer
273 views

How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
2
votes
3answers
67 views

solving a second order nonlinear pde

I would like to solve the following PDE, $$f_{y}^{2} = 2 f f_{yy}$$ where $f= f(x,y)$ is a real function of two variables $x,y$. My solution : derivative of $f_{y}^{2}$ with respect to $y$ is itself, ...
0
votes
1answer
115 views

Runge-Kutta force at each time-step

Consider that I am solving a second order ODE using RK2/RK4. The ODE represents simple equations of motion: Equations of motion I am trying to solve: \begin{align} \frac{dx}{dt} &= v \\[.3em] ...
1
vote
2answers
61 views

Differential and Differential Equation - Difference in meaning?

I am a little confused, an exercise by a teacher has been set which says: For $X_t = 2e^{B_t}$ Where $B_t$ is brownian motion at time $t$. a) Find the stochastic differential $d(X_t)$ b) Find the ...
0
votes
0answers
74 views

Solve $y'' + \epsilon y' + 1 = 0$ with initial conditions $y(0) = 0$ and $y'(0) = 1$

Let $\epsilon << 1$. I guess I'm trying to use perturbation method but I've been getting really weird numbers when I'm determining the initial conditions. Can someone perhaps help me with ...
1
vote
0answers
35 views

Solving Differential Equation Multiple Ways

I am currently self learning differential equations and I use the book Elementary Differential Equations. My question is that I saw many ways to solve a DE. Can I use any method to solve any DE? For ...
1
vote
1answer
60 views

Linear differential equation

In this linear differential equation, should I eliminate the $\tan x$ in the expression in order to get $\frac yx$ or may I cancel $\tan x$ by $\tan^2x$? $$\frac{dy}{dx} = \tan x y + \cos x$$
0
votes
1answer
32 views

2nd Order ODE: Variation of Parameters

I used Abel's theorem $W=ce^{-\int p(t) dt}$ where in this case $p(t)=0$ so the wronskian is a constant. There are 2 ways I know of for variation of parameters. One is where you know that $y_1$ ...
2
votes
3answers
127 views

How to solve this differential equation, involving leibniz notation?

I thought I was pretty good at calculus, but this one has stumped me. I can do many almost identical examples, but I can't seem to extrapolate the skills needed to this one tricky problem. $${dy ...
1
vote
0answers
40 views

Solving $\nabla^2U(x,y)=0$ on a donut with two inhomogeneous boundary conditions

I am given $\nabla^2U(x,y)=0$ on a donut-shaped region, with the inner circle being of radius $r_1$, and the outer circle $r_2$. In polar coordinates, the relation is ...
0
votes
1answer
60 views

Differential equation (Stationary Point)

Find the general solution to the differential equation $$x\frac{dy}{dx}-y-2x^2+1=0$$, expressing y in terms of x. Find the particular solution which has a stationary point on the positive x-axis. ...