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

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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 ...
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1answer
16 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
9 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 ...
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1answer
22 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 ...
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2answers
26 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
9 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
16 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 ...
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1answer
24 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
30 views

Help solving differential equations

I have a few questions. I am trying to teach myself differential equations and would like to know how to classify and solve the following equations: $y''+ 4y'+5y=2e^{-2x}cos(x)$. I know this is a ...
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1answer
45 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
25 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 ...
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1answer
33 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
46 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
21 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 ...
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1answer
20 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 ...
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1answer
20 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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0answers
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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
47 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
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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
23 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 ...
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2answers
45 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: ...
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3answers
36 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} ...
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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
19 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
16 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
14 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
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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] ...
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1answer
48 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
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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 ...
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1answer
13 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, ...
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0answers
11 views

splitting a system of ODEs into linear constraints and a smaller system using matrix Null Space

This problem originates from chemistry. Let us assume we want to solve a system of ODEs describing the evolution of the concentrations of the species in a chemical system with n species and k kinetic ...
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3answers
40 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, ...
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1answer
32 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] ...
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0answers
16 views

Can someone check my work on the PDE five-point scheme problem?

I'm working on a practice exam for an upcoming final exam next week, but unfortunately the professor did not release solutions to the practice exam. I was hoping someone here could verify that I did ...
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2answers
42 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 ...
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1answer
37 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 ...
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0answers
21 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 ...
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0answers
27 views

Phase Diagram Bifurcations and much more. [on hold]

this is a problem I came across while solving a paper. A temporary guide would be appreciated, because I just need a kick-start,I'm not familiar how I should be tackling the question, so a guide on ...
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0answers
13 views

Use a Lie series in order to find the solution to initial value problem

We were presented with a fairly difficult bonus question on my multivariable calculus exam today. I was hoping you all could hope me crack it. The question is as follows: Use a Lie series to find, ...
1
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1answer
37 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 $\tan^2x$? $$\frac{dy}{dx} = \tan x y + \cos x$$
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1answer
13 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 $y_1$ ...
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0answers
17 views

Convergence of the Midpoint (Leapfrog) method when applied to $u'(t)=\lambda u(t)$?

So, I am trying to solve this question: where example 7.7 can be found here: http://i.stack.imgur.com/PVCIC.png My approach: Forward Euler (FE) method is given by: ...
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3answers
75 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 ...