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

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6
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0answers
25 views

Frobenius method, why is it an issue when the roots of the indicial equation differ by an integer

When solving second-order differential equations by the Frobenius method at a regular singular point, you are supposed to use the two roots of the indicial equation to give you two independent ...
2
votes
0answers
8 views

Academic Prerequisite of Dynamical system and applied PDE

With a very strong intention on future research closely related to Dynamical Systems and applied PDE. What are the materials as a prerequisites which are strongly recommended to study hard during ...
0
votes
1answer
20 views

What is the Jacobian of the following function

Consider a function F: $R^n \to R^n$ defined by $$f(u) = A*u*(n+1)+\lambda *B$$ Where A is a tridiagonal n-by-n matrix with -2 on the main diagonal and 1 on the off diagonals. B = $\begin{pmatrix} { ...
1
vote
2answers
25 views

Solving an ordinary differential equation with initial conditions

Can someone please help me with this ODE problem? Here is the question: Consider the ODE $ {d^2 U\over dx^2} - [{s^2\over c^2}]U=e^{{-sx\over v}}. U(0) = 0, U(x)$ is bounded as $x$ goes to ...
0
votes
1answer
16 views

What is the difference between single and double modulus signs. Do they both mean magnitude?

What's the difference between a set of single modulus and a set of double modulus signs? On textbooks I have seen the magnitude of two vectors vector as |x-y| but I've seen other sites where they've ...
0
votes
0answers
26 views

Find the Jacobian of the following function

We start with a problem where we need to find u such that $$\frac { { d }^{ 2 }u }{ dx^{ 2 } } +\lambda { e }^{ u }=0$$ with 0 < x < 1, u(0)=u(1)=0 and $\lambda$ is a known constant. We ...
2
votes
1answer
40 views

Solving weak 2 body problem

I tried to solve a physics problem about two body problem where the masses $M$ and $m$ are $M \gg m$. The body $m$ is at radius $R$ from the mass $M$ and is falling down with initial speed $v(0) = 0$. ...
0
votes
2answers
24 views

What happens when the integrating factor is a factor of neither x or y?

I've got a differential equation $2\sin(\omega y)dx+\omega\cos(\omega y)dy=0$ They are not exact equations so I need an integrating factor to make them so. I am loosely following this website as a ...
2
votes
0answers
20 views

Derivative of terminal state w.r.t. the inital conditions.

Let $x\in R^n$ and consider the system $$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$ and suppose that we know it's exact or very accurate solution $x(t)$ for the time interval $[0,T]$. I'm ...
3
votes
3answers
80 views

how to find matrix from its exponential form

I know about the relation $$\frac{d}{dt}e^{At}=Ae^{At}$$ Is the only way to use it is to find the inverse of $e^{At}$ and then post-multiply on both sides? Is there a better approach?
0
votes
0answers
23 views

system of two ODE: plot $x$ versus $t$

I have an assignment in which I have a system of two ODEs for $x(t)$ and $y(t)$. I'm asked to find critical points, draw phase portraits, etc. Once of the questions asks to draw a plot of $x$ versus ...
0
votes
0answers
11 views

coupled heat transfer equation

I want to try to solve a strong coupling problem, I have a variable $\zeta as$ : \begin{equation} \zeta(x,y,T)=\frac{\frac{R(x,y,T)}{\sqrt{2}}-F(T)}{F(T)-E(T)} \end{equation} Where F(T) and E(T) are ...
0
votes
1answer
40 views

Solving for Center Manifold with Parameter

I have a system of ODEs given by $$\frac{dX}{d\tau}=\beta X\left(1 - \frac{X+Y}{N}\right)$$ $$\frac{dY}{d\tau}= Y\left(1 - \frac{X+Y}{N}\right)$$ where $\beta $ is a parameter. How should I ...
0
votes
0answers
22 views

Can the transition probabilities of an inhomogeneous Markov chain be written as an exponential?

If $Z_t$ is a homogeneous continuous-time Markov chain with finite state space $E=\{1,\ldots,p\}$, transition matrices $(P(t))$ and intensity matrix $Q$, it holds that $$ P(t) = \exp(tQ), $$ see for ...
0
votes
0answers
14 views

Matrix multiplier for ODE

I have matrix C with dimensions $3 \times 3 $ and it is skew symmetric too C is given by $C(0,0)=0,C(1,1)=0,C(2,2)=0 \tag 1$ $C(1,0)= sc_0+ px (c_1-c_0),C(0,1)=-C(1,0) \tag 2 $ $C(0,2)= ...
4
votes
3answers
102 views

General solution of $\frac{\partial^2}{\partial t^2} x(t) + \omega^2 x(t) = 0$

Consider $$\frac{\partial^2}{\partial t^2} x(t) + \omega^2 x(t) = 0$$ 1) Show that $\left(\frac{\partial x}{\partial t}\right)^2 + \omega^2 x^2$ is constant in $t$, and 2) deduce that the general ...
1
vote
1answer
54 views

Solving a master equation with linear coefficients

I have the following PDE: $$ \partial_t P(x,y,t)=x\partial_xP(x,y,t)+(y-1)\partial_yP(x,y,t)+2P(x,y,t). $$ Mathematica suggests that the solution is $$ ...
0
votes
1answer
33 views

Continuity of $K(x,y)$ satisfying $g(x)= \int_0^1 \! K(x,y) f(y)\ \mathrm{d}x $ and $ \frac{d^3g}{dx^3} = f$

$g(x)$ is defined as the following : $$g(x)= \int_0^1 \! K(x,y) f(y)\ \mathrm{d}x $$ where $K(x,y)$ is continuous in $ 0 \leq x \leq 1 $ , $ 0 \leq y \leq 1 $, and $f(x)$ is continuous in $ 0 \leq x ...
9
votes
4answers
135 views

Solution to $y(x) + y'(x) + y''(x) + y'''(x) + \cdots = 0$

Is there a non-trivial solution to the following differential equation? $$y(x) + y'(x) + y''(x) + y'''(x) + \cdots= 0$$ That is, is there a smooth function $y : \mathbb{R} \to \mathbb{R}$ such that ...
2
votes
0answers
52 views

Ordinary differential equation­

$$\dfrac{dy}{dx}-\dfrac{\tan y}{1+x}=(1+x)e^x\sin y$$ I tried $\sin y=t$ but failed. It seems to immune to methods I know of or I am just unable to make the right substitution... Wolfram alpha ...
2
votes
0answers
20 views

How to use the Mehler kernel to get the solution of the Quantum harmonic oscillator with a given initial condition

In this wiki-article http://en.wikipedia.org/wiki/Mehler_kernel the fundamental solution of the differential equation for the Quantum harmonic oscillator is provided by the Mehler Kernel: ...
0
votes
0answers
12 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
0
votes
2answers
23 views

Solution of the Legendre's ODE using Frobenius Method

This is the Legendre's differential equation given in my book: $(1-x)^{2}\ddot{y}-2x\dot{y}+k(k+1)y=0$ I solved this equation by taking: $y=x^{c}\{a_{0}+a_{1}x+a_{2}x^{2}+.....+a_{r}x^{r}+.....\}$ ...
0
votes
0answers
13 views

obtain the continuous form of a master equation

I have the following master equation: $$ \partial_tP(x,y,t)=-[x+(1-x-y)]P(x,y,t)\\ +(x+\epsilon)P(x+\epsilon,y-\epsilon,t)\\ +(1-x-y+\epsilon)P(x,y-\epsilon,t) $$ where $\epsilon$ is a small number, ...
-2
votes
1answer
20 views

Potential equations with boundary conditions [on hold]

I'm stuck on this problem. If you could help me solve it by explaining it in easy to understand steps, I'd grateful! Thanks!
1
vote
0answers
32 views

best introductory intuitive books for learning ODE

I want to know best introductory intuitive books for learning ODE (mainly interested in Picard' theorem, Gronowall's inequality and most importantly stability). I started with Philip Hartman. Not ...
2
votes
2answers
101 views

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don't have $2x$ but just $x$. ...
3
votes
1answer
31 views

Oscillations about equilibrium for coupled differentail equations

I have the following system of equations: $$\begin{align} \frac{dX}{dt} &= 2Y-2\\ \frac{dY}{dt} &= 9X-X^3 \end{align}$$ I would like to study the property of solutions to this function about ...
3
votes
2answers
88 views

Solving $\frac{d f(x)}{dx} + f(x-1) = x^2$

Given following differential equation: $$\frac{d f(x)}{dx} + f(x-1) = x^2$$ where $ f(x)=0 $ for $x \leq 0 $. How do I find the solution for $ x \geq 0 $ ? I understand that for $ 0 \leq x \leq 1 ...
-3
votes
2answers
29 views

differential equation population problem [on hold]

Consider an initial population of 1000 field mice that grows at a rate proportional to the current population p, so that dp/dt=kp A- What is the IVP the models this scenario? that is, give the ...
-2
votes
1answer
19 views

differntiability of the following function [on hold]

Let f(x)=sinx/x,x≠0 =1 ,x=0, then f is a)discontinuous b)continuous but not differentiable c)differentiable only once d)differentiable more than once
0
votes
1answer
52 views

Doubt on an ODE problem

Consider the following differential equation $$x'(t) = h(x(t))$$ Consider a function $x(t)$ which satisfies the differential equation for $0 \lt t \leq 1$ and another function $y(t)$ for $0.5 \leq ...
1
vote
1answer
41 views

Solution to diff eq

Check whether the function $y=\sin(3x)/3$ is a solution of $xy'+y+3\cos3x$ with the initial condition $y(\pi)=0$ Find $xy'$ for the function $y=\sin(3x)/3$ I am a ex-math minor who is just trying ...
1
vote
1answer
39 views

Solve the equation $3xy''+5y'+3y=0$

For the equation $$3xy''+5y'+3y=0$$ i have to find two independent solutions for $x>0$. And i have to see if this solutions are analytic at 0. My approach: I try to solve this problem using ...
2
votes
0answers
45 views

Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
0
votes
0answers
23 views

Question on linear partial difference equation with three independent variables $n$, $m$, $k$.

Would it be possible to find closed form for the recursively defined algebraic function of 3 integer arguments F[n,m,k] ? Here are the details: Recursion definition is ...
0
votes
0answers
22 views

Solving system of equations

I have the following set of equations: $y = f(a,b)$ $a = f(y)$ $\dot{b} = f(b,y,\dot{y})$ which I like to solve for $y$. I was wondering if there is some numerical method which I can apply to ...
1
vote
2answers
20 views

What does a number in gradient symbol subscript means?

While solving some problems I have encountered a subscript in front of a gradient symbol. I'm unable to understand it, I know a superscript of 2 on gradient symbol means Laplacian but what does ...
1
vote
0answers
31 views

Summation of Recurrence (Convergent series)

I have solved this issue. Would you please verify whether I am correct or not? Motivation for the post is our previous discussion link.I am restating my problem with additional elaborated explanation ...
0
votes
0answers
18 views

4th order method

I am asked to solve a ODE using the 4th order Runge-Kutta method, and then given the analytical answer, 'show the method is 4th order numerically' . What does the question 'show the method is 4th ...
0
votes
0answers
15 views

How can I check Lipschitz condition for this function?

I have system of ODE that have these equations: $\partial_t f_1(t,u) = 2 \int_0^1 f_1(x,t) dt + f_1(u,t) \int_0^1 f_2(x,t) dx +f_1(u,t) \int_0^1 f_2(x,t) dx + 8(2u \int_0^1 f_1(x,t) dx - u^2 ...
0
votes
1answer
16 views

Using differentials, estimate the difference in the deflection between the point midway on the beam and the point 1 10 ft above it

So I've been trying to figure out the problem for about an hour and I cannot figure it out. Question: To study the effect an earthquake has on a structure, engineers look at the way a beam bends when ...
1
vote
1answer
47 views

A basic confusion in the proof of Picard's existence theorem

In the proof of Picard's existence theorem of solution of ODE I don't understand the following step: Once it proves that the limit of uniformly convergent series is a continuous function then it ...
-2
votes
1answer
25 views

Euler's method problem [on hold]

Use one iteration of Euler's Method with step size of $h=1$ to approximate the solution to the differential equation at $t=1$: $\begin{array}{l}\frac{{dy}}{{dt}} = {y^2} + t - 1\\y(0) = - ...
-1
votes
1answer
30 views

Is that equation an exact value for differential equation [on hold]

Is $2xy^2+4-2(3-x^y)\frac{dy}{dx}=0$ an exact equation? Justify your answer B- solve $2xy^2+4-2(3-x^y)\frac{dy}{dx} = 0$ simplicity.
1
vote
2answers
32 views

Newton's law of cooling problem Differential Equation

Suppose that a building loses heat in accordance with Newton's law of cooling which state the rate of change of temperature within the building is proportional to the difference inn the inside ...
1
vote
2answers
25 views

tank problem Differential equation

A tank is partially filled with 100 gallons of coffee in which 10 lbs of sugar is dissolved. Coffee containing 1/3 lb of sugar per gallon is pumped into the tank at rate 3 gal/min. The yummy ...
0
votes
0answers
26 views

What are the equilibrium solution [on hold]

Given $\frac{dy}{dx}= y(y+2)(y-3)^2$ $a-$ what are the equilibrium solution for $\frac{dy}{dx}= y(y+2)(y-3)^2$ $b-$ sketch the phase line and classify all equilibrium pointe $c-$ Next to the phase ...
1
vote
0answers
43 views

Insightful books on differential equations?

What are some recommendations for insightful books on differential equations and difference equations? These books don't need to be in the format of a textbook, and don't need to provide the same ...
2
votes
2answers
55 views

Separation of variables: when to have exponential solution and when sinusoidal?

In separation of variables, one can assume a solution of V(x,y) = X(x)Y(y) and after plugging this into Laplace's equation which is: ${{\partial^2 V} \over {\partial x^2}}$ + ${{\partial^2 V} \over ...