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

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1answer
16 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 ...
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0answers
7 views

Solution space of Linear homogeneous differential equation

The solution space of a L.H.D.E of order n is a vector space spanned by n base vectors, right? So any solution is then a vector of the solution space -> a linear combination of the base vectors. But ...
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0answers
40 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 ...
3
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0answers
13 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 ...
3
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0answers
18 views

Three-Variable Systems of Differential Equation s

I understand fully how to solve two systems of differential equations for both linear, nonlinear, homogenous, and nonhomogenous using linearization and eigenvalue\Liapunov equilibria analysis. Such ...
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1answer
22 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, ...
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1answer
26 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 ...
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2answers
47 views

differential equation question $\frac{dy}{dx} = \frac{2xy}{x^2 + y^2}$

how do you solve this ? $$\frac{dy}{dx} = \frac{2xy}{x^2 + y^2}$$ thank you in advance!
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0answers
18 views

Show that the solution of the Cauchy problem $x(t,t_0,x_0)$, $x(t_0)=x_0$ is definite for all $t\geq t_0$.

Consider the system: $$x' = A(t)x + b(t)$$ where matrices $A(t)$ and $b(t)$ are only integrable on compact sets of $\mathbb{R}$. Show that the solution of the Cauchy problem $x(t,t_0,x_0)$ is ...
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0answers
14 views

the Fisher equation has no positive traveling wave solution

Use the linearization method to prove that for any $c\in(0,2)$,the Fisher equation$u_t=u_{xx}+u(1-u)$has no positive traveling wave solution $U(x+ct)$ with $U(-\infty)=0$
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0answers
13 views

Use the persistence theory to find a set of sufficient conditions for two species competitive ODE system

Use the persistence theory to find a set of sufficient conditions for two species competitive ODE system $$\frac{du_1}{dt}=u_1(b_1-a_{11}u_1-a_{12}u_2)$$ ...
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2answers
54 views

Can somebody please show me the necessary steps to solve this Calculus problem?

I have a homework assignment that asks me to solve the differential equations and it gives me: \begin{align*} xy^2y' & = 2-x\\ y''+4y & = 8x\\ y(1)& =1 \end{align*} Are these three ...
2
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1answer
24 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 ...
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2answers
8 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
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1answer
21 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 ...
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0answers
13 views

Second order linear ODE and undamped

I am a bit confused with this problem: An object with mass 1 slug is attached to a vertical coil spring of spring constant of 1 pound per foot. After coming to equilibrium, the object is set into ...
1
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1answer
25 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
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2answers
30 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.
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0answers
12 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 ...
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0answers
18 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
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1answer
26 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 ...
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1answer
35 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?
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1answer
47 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) ...
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0answers
32 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 ...
3
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1answer
36 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 ...
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0answers
48 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 ...
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1answer
23 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 $B_R:=B_R(0)$ ...
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0answers
10 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
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1answer
22 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 ...
6
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2answers
152 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
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1answer
21 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.
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9 views

Evaluating vorticity as a function of velocity components.

So i have the following question.. Consider the axisymmetric flow of a viscous fluid u = ($ \frac{-\alpha r}{2} $, v(r), $\alpha z$) in cylindrical polar coordinates, where $\alpha$ is a positive ...
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2answers
48 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 ...
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1answer
32 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 ...
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0answers
19 views

Step Function Section 6.3 [on hold]

Could anyone help me write the function in terms of unit step function? $$ f(t) = \begin{cases} -5, & 0 \leq t < 1\\ 4, & 1 \leq t < 5\\ -3, & t \geq 5 \end{cases}$$
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0answers
13 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 ...
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0answers
24 views

How can I solve differential equation near point that is not normal

Let we have the following differential equation : $$2z(z+1)w''+z(z+1)w'-w=0$$ By power series near the point $z_0=0$ the problem that the point $z_0$ isn't normal point for this equation , so how can ...
0
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2answers
50 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
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3answers
37 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
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2answers
65 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 ...
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2answers
42 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 ...
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2answers
20 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 ...
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1answer
30 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 $$ ...
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1answer
17 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 ...
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1answer
15 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 ...
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2answers
34 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 ...
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0answers
27 views

Linear high-order ODEs

I'm looking for an approximate solution of this ODE: \begin{equation} \left(\frac{a_7}{x^6}+\frac{a_8}{x^4}\right)y+\left(\frac{a_9}{x^5}-\frac{a_{10}}{x^3}\right) ...
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0answers
12 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] ...
4
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1answer
53 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 ...
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0answers
10 views

A collective spin motion, related to differential equations. - - how to prove y component of the field is zero throughout the motion?

This is a pure mathematical question, here is a little background for the interested reader, you can jump directly to the mathematical part if you are not interested. background Imagine we have ...