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

learn more… | top users | synonyms (1)

6
votes
3answers
232 views

Solutions of homogeneous linear differential equations are a special case of structure theorem for f.g. modules over a PID

In this other post, Qiaochu Yuan comments that the solutions for the homegeneous linear differential equation with constant coefficients are a special case of the structure theorem for finitely ...
5
votes
2answers
932 views

Cleaning up the Great Lakes - Differential equation systems

Problem Details The idea of the problem is to find out how long it would take to flush the Great Lakes of pollution. They're set up as a series of five tanks and you are given inflow rates of clean ...
5
votes
1answer
86 views

Decomposite a vector field into two parts

Let A be a region in $\mathbb R^3$, and suppose $ \vec {\mathbf F}$ is a smooth vector field on A. I was asked to show that I can write $\vec {\mathbf F}=\vec {\mathbf F_1}+\vec {\mathbf F_2}$, s.t. ...
4
votes
6answers
1k views

ODE introduction textbook

Unfortunately I have reached the maximum number of math classes I can take for my undergraduate degree. I still wish to study basic ODEs and basic number theory. What is a good textbook with an ...
3
votes
1answer
41 views

Uniqueness of solutions to linear recurrence relations

I understand that if I have a linear homogeneous recurrence relation of the form $q_n = c_1 q_{n-1} + c_2 q_{n-2} + \cdots + c_d q_{n-d}$, I can construct the characteristic polynomial $p(t) = t^d - ...
3
votes
1answer
3k views

Polar coordinates differential equation

I have the following ODE: $$\dot x=-y(x^2+y^2), \dot y=x(x^2+y^2)$$ I want to sketch the phase portrait (manually) and I want to find the flow $\phi_t$, the orbit $O(x_0)$ and the limit set ...
3
votes
2answers
411 views

Advection Diffusion Equation on Semi-Infinite Domain

Regarding the BVP $$u_t(x,t) - v u_x(x,t)=k u_{xx}(x,t),\qquad x\geq0$$ with BC $u_x(0, t)=0$ for $t\geq 0$, and parameters $v,k>0$, I have some questions. Does an expression for the Green's ...
3
votes
1answer
531 views

Explain the error term in Euler method

Task: I had to find out some estimates for M and L to make sure the proportional accucrazy is not above $10^{-4}$ in the Euler method with the problem below. I am trying to understand the page 672 on ...
2
votes
2answers
446 views

How to solve this ODE?

For a certain problem, I am trying to solve the ODE $$\ddot{z}(t) - \omega^2 z(t) = f_0 \Big(e^{-i(\omega+\delta)t}+e^{-i(\omega-\delta)t}\Big)$$ where $\omega$ is a real and $\delta$ is close to ...
1
vote
2answers
94 views

Suggestion for a Lyapunov function

Consider the differential system $$ x'=x+y $$ $$y'=x-y+xy$$ What would be a Lyapunov function for this system at $(0,0)$? I have considered functions $V(x,y)=ax^{2n}+by^{2m}$ but none of ...
1
vote
1answer
2k views

Pursuit Curve. Dog Chases Rabbit. Calculus 4.

(a) In Example 1.21, assume that $a$ is less than $b$ (so that $k$ is less than $1$) and find $y$ as a function of $x$. How far does the rabbit run before the dog catches him? (b) Assume now that ...
0
votes
1answer
114 views

Questions about the hyperbolic system of equations

$$u_t+A(x,t,u)u_x=b(x,t,u) \tag 1$$ $$u=(u_1, \dots, u_n), b=(b_1, \dots, b_n)$$ $$A=[a_{ij}], i,j = 1, \dots, n$$ $$$$ We set the question if there are characteristic directions at the path of which ...
8
votes
2answers
210 views

A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here ...
8
votes
1answer
410 views

To get addition formula of $\tan (x)$ via analytic methods

Assume that we only know $\tan (0)=0$ and also given the relation $\tan'(x)=1+\tan^2(x)$ about $\tan (x)$ and we do not know other $\tan (x)$ relations of trigonometry. How can I get the additon ...
8
votes
1answer
171 views

Solving the differential equation $\frac{dy}{dx}=\frac{3x+4y+7}{x-2y-11}$

How do we solve the differential equation $$\frac{dy}{dx}=\frac{3x+4y+7}{x-2y-11}$$? I tried substituting $v=yx$ but I do not seem to be getting anywhere.Putting $u=x-2y$ yielded nothing better. ...
7
votes
1answer
95 views

Solve $y' + \frac1y + \frac1x =0$ Differential Equation

Do you have any suggestions for how to sole this differential equation? $y'+\frac1x + \frac1y =0$ ? :) I tried solving this by changing variable in the form of $v=x^\alpha*y^\beta$ but it didn't ...
7
votes
1answer
131 views

Periodic solution of differential equation y′=f(y)

Let $f∈C^∞(ℝ^2,ℝ^2)$ and $∀x∈ℝ^2$ $f(kx)=k^2f(x)$ for $k∈ℝ$ Show that any periodic solution of $y′=f(y)$ is constant. My attempt : Let $\lambda \in \mathbb{R}$. Let $g$ a periodic solution ...
7
votes
1answer
393 views

ODE Laplace Transforms: what impulse brings an oscillating system to rest?

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
7
votes
1answer
2k views

Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
6
votes
2answers
122 views

Geometric series of an operator

In solving a first order linear differential equation $(1-D)y=x^2$ where $D\equiv \frac{d}{dx}$ the way I learnt was that we proceed as ...
6
votes
2answers
767 views

General solution for $y^{iv}+ 2y''+y=\cos x$

Here is another problem from Mathews and Walker that has given me some trouble. 1-18. Find the general solution of $y^{iv}+ 2y''+y=\cos x$. Note: Thanks, everyone, for clearing up the ...
5
votes
1answer
94 views

projectile motion (with height) complicated

When a child standing on a horizontal path throws a ball, it leaves her hand from a point that is 90cm vertically above the path. The ball clears a 4.5 m high wall that is 10.5 m away from ...
4
votes
0answers
85 views

Green's function in a moving frame for a constant heat source

I am looking for the Green's function of the problem in two dimensions $r =(x,z)$, \begin{equation} \nabla^2g + \frac{v}{D}\frac{\partial g}{\partial z} = -\delta (r-r_0) \end{equation} Which ...
4
votes
2answers
109 views

Limit of solution of linear system of ODEs as $t\to \infty$

I am completely stuck on the following problem: Consider the linear system: $x'(t)=A(t)x(t)$ where $A(t)$ is an $n$ by $n$ matrix. Assume that $\lim_{t\to \infty}A(t)=B$. Suppose that each eigenvalue ...
4
votes
1answer
292 views

Background for studying and understanding Stochastic differential equations

Assume I have back ground of the following knowledge based on the textbook as : ODE : ODE by Tenenbaum Baby probability : Ross 's baby probability Baby real anlysis : Bartle's introduction to real ...
4
votes
1answer
171 views

Some Scaling Estimate for Heat Kernel

NOTE. I have rewritten the question to summarize my current progress on this question. The bounty is for completing what I have done so far, or by offering a more elegant solution probably based on ...
4
votes
1answer
528 views

The characteristic polynomial of a recurrence relation.

If I have a linear homogeneous recurrence relation $$y_n=c_1y_{n-1}+\ldots+c_ky_{n-k},$$ I can get its characteristic equation, which is $$r^k=c_1r^{k-1}+\ldots+c_k.$$ In particular for ...
4
votes
2answers
2k views

Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...
4
votes
1answer
113 views

Special Differential Equation (continued-2)

May you help me out in solving inhomogeneous differential equation looking like[this is radial part of Schrodinger equation]: ...
4
votes
4answers
1k views

Help with solving differential Equation using Exact Equation method

I need to learn how to solve differential equations using either the Exact Equation Approach and or the Special Integrating Factor methods. Below is a differential Equation to solve. $(2xy^2 + \cos ...
3
votes
1answer
105 views

Energy Method to show uniqueness of solution of PDE

In my notes there is the following example about the energy method. $$u_{tt}(x, t)-u_{xxtt}(x, t)-u_{xx}(x, t)=0, 0<x<1, t>0 \\ u(x, 0)=0 \\ u_t(x, 0)=0 \\ u_x(0, t)=0 \\ u_x(1, t)=0$$ ...
3
votes
3answers
155 views

Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-1$

For this problem, $y$ is a function of two variables: one space variable $x$ and one time variable $t$. $k > 0$ is some constant. And $x$ takes is value in the interval $[0, 1]$ and $t \ge 0$. At ...
3
votes
1answer
3k views

Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order ...
3
votes
0answers
152 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...
3
votes
4answers
1k views

particular solution of $4y''-y= \sin(x)\cdot \cos(x/2)$

So I'm working with a nonhomogeneous second order differential equation: $$4y''-y=\sin(x)\cos(x/2).$$ I know that the general solution, $y$, equals $y_c + y_p$ where $y_c$ is the general ...
2
votes
1answer
85 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
1answer
126 views

Counter-example to Cauchy-Peano-Arzela theorem

I was looking for a counter-example to Cauchy-Peano-Arzela theorem. I found this paper (in french) from Dieudonné. [acta.fyx.hu] Take $E = c_0$ to be the space of real sequences converging to $0$, ...
2
votes
0answers
131 views

perturbation question

I'm a little stuck with a problem and I was hoping that you guys could help. Question: A projectile is fired up from the earth with an initial velocity of $v_0$ upwards. Accounting for air resistance, ...
2
votes
1answer
279 views

How to get the linear equation system for finite element method from the variational formulation

Let the problem be $$-u'' + a(x) u = f , \;x \in \Omega = ]0,1[ , \;u(0) = \alpha ; \;u(1) = \beta,$$ where $f \in L^2(\Omega) , a(x) \geq a_0 > 0 , a(x) \in L^{\infty}(\Omega).$ This problem ...
2
votes
2answers
336 views

resources to study PDE from

I am an undergrad engineering student. I recently completed my second year, with that said, I have taken several calculus courses. Most recently I completed differential equations and multivariable ...
2
votes
2answers
140 views

Stability analysis, or, Can we prove this limit to be zero?

Let's think about this ODE $$ \dot{y}(t) = \gamma \left(g(t) - y(t)\right),\quad \gamma > 0, $$ where $g(t)$ is a Lipschitz continuous function. It can be seen that the value of $y(\cdot)$ goes ...
2
votes
2answers
2k views

second derivative of the inverse function

I know that the derivative of the inverse function of $f(x)$ is $g'(y) = \frac{1}{f'(x)}$ But how to derive the formula for the second derivative of g(y) knowing that $\left[\frac{1}{f(x)}\right]' = ...
2
votes
3answers
2k views

Cat Dog problem using integration

Consider this equation : $$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$ where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$. Now how to proceed ? This equation ...
1
vote
1answer
32 views

Show solution to character identity

In my analysis textbook in a chapter on fourier series it says that it is easy to show the following: Any complex-valued $C^1$ solution to the character identity must be of the form $e^{\lambda x}$ ...
1
vote
1answer
30 views

Converting an ODE in polar form

Convert the ODE system $$ \dot{x}=\begin{pmatrix}a(t) & b(t)\\c(t) & d(t)\end{pmatrix}x $$ into polar form. You should get two equations $$ \frac{d}{dt}\Phi(t)=...\\ ...
1
vote
1answer
87 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
1
vote
1answer
63 views

Linear equation solving

I cant solve $tx'+\dfrac{tx}{\sqrt{1+t^3}}=1$ I have tried to do it like an homogenian but i cant integrate $\dfrac{1}{\sqrt{1+t^3}}$ so i suposse it must be done by another method
1
vote
2answers
247 views

Understanding differentials

What is a good reference to learn about differentials and related topics. Some of my questions are: Why is it possible to split $dy/dx$ into individual terms $dx$ and $dy$? In a separated ...
1
vote
2answers
315 views

Comparison theorem for systems of ODE

Let vector-function $x(t)$ satisfy a differential equation $$ \dot x = f(x), $$ and a vector-function $y($t) satisfy a differential inequality $$ \dot y \leq f(y) $$ with starting positions $y(0) ...
1
vote
2answers
121 views

Easy way to solve this non-linear second degree DY $\sqrt{y}\;y''=1$?

I prooceeded by integrating both sides $$y'=\int y^{-\frac{1}{2}} dx=\cdots$$ so I got $(y')^{2}+C y' - \frac{1}{2} \sqrt{y} = 0$ but I am thinking that I am proceeding the wrong or the ...