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

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3answers
40 views

Why does my derivation of $\mathcal{L(\frac{f(t)}{t})}$ lead to a wrong answer?

I'm trying to prove that $$\mathcal{L(\frac{f(t)}{t})(s)} = \int_s^{\infty}\mathcal{L(f(t))}(u)du$$ Here's my attempt: $$\mathcal{L(\frac{f(t)}{t})}(s)=\int_{0}^{\infty} \frac{f(t)}{t}e^{-st}dt$$ ...
0
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0answers
20 views

Finding alternating series for Power series

Given data and conditions I have a power series, $PS(x) = \sum_{n=0}^\infty R_nx^n$. I have a infinite GP,something like G(x) = $\sum_{k=0}^\infty ax^k = \frac{a}{1-x} $ . Never take G(x),such ...
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0answers
14 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
1
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3answers
25 views

Linearize a first order differential equation

The system described by $x'=2x^2-8$ is linearized about the equilibrium point -2. What is the resulting linearized equation? Answer is $x'=-8x-16$. How? I have no idea how it went from the first ...
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1answer
15 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < ...
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1answer
28 views

Ricatti differential equation solution

I attempting to solve some Riccati differential equations. It has been a while since I have worked with differential equations so I am rusty. I would appreciate if someone would show me how to do ...
1
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2answers
33 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
2
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1answer
35 views

Solving $xy''-(1-x)y'+y=0$

$$xy''-(1-x)y'+y=0$$ So I know how to solve this via power series. Recently, a friend of mine was asking me how one could solve this without using series. I've got no real idea how to answer this ...
0
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1answer
36 views

Prove two solutions of differential equation are the same

In a recent work I had to solve the following differential equation: $$ r x''(r)+r x'(r)^2+x'(r)-\frac{4}{r}=0~~. $$ To do so I used two methods and I got, using each, two solutions with different ...
0
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1answer
61 views

solving the equation

let there be a function $ f(x)= \ln x-kx^2, k>0$ determine for whihc values of $ k$ ,the equation $f(x)=0.5$ has a single solution; attemp to solve: $$0.5 = \ln x-kx^2$$ $$kx^2 +0.5 = \ln x $$ ...
1
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1answer
32 views

Find an expression of the direction field

I have a directions vector field which I got empirically using quiver in Matlab. I want to find some analytical expressions that might work at least in part of the direction field. How can I ...
0
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0answers
14 views

Accuracy of a finite-difference method for numerically solve a PDE or BVP

When solving the Poisson Equation $$-u''(x)=f(x)$$ with Dirichlet-Neuman boundary conditions $$u(0)=0, u'(1)=0$$ using a finite difference 2-order centered scheme and a 2-order upwind ...
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0answers
9 views

Potential equation in rectangle with boundary values

I'm running into problem with the boundary conditions for u(x). I get u(x) = sin((n*pi*x)/a) based on u(0,y)=0, but that doesn't agree with du/dx(a,y)=0 unless the whole function u(x)=0. Is that the ...
0
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2answers
17 views

General Solution - Differential Equation

Question asks to find the general solution of the differential equation. $$\frac{1}{r}\left(\frac{d}{dr}\left(r\frac{dw}{dr}\right)\right)-\frac{\lambda^2}{r^2}w=0.$$ The answer given is ...
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0answers
23 views

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the green's function. I have to solve the given differential equation using Green's function method $\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta ...
1
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1answer
15 views

Second order D.E. - general solution

If $y=y(x)$, and we have the differential equation $y''=-k^2y$ for some constant $k>0$, then wolfram alpha gives the general solution as $y=A\cos(kx)+B\sin(kx)$. I've also seen this result used ...
2
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2answers
61 views

How to prove that solution to ODE in spherical coordinate is equivalent to the ODE in cartesian coordinates if it is a thin shell

Solving a diffusion-type ODE across a spherical shell, the equation is: $$\frac{d}{dr}\left(r^2\frac{df}{dr}\right)=0\tag{1}$$ with boundary conditions $f(r_1)=f_1$ and $f(r_2)=f_2$. The solution is: ...
2
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3answers
32 views

Differential equations with missing variable

$$y\cdot y'' + (y')^2 = 0$$ I'll make $V=\frac{\mathrm{d}y}{\mathrm{d}t}$, $y''=\frac{\mathrm{d}v}{\mathrm{d}t}=V\cdot\frac{\mathrm{d}v}{\mathrm{d}y}$ $$\Rightarrow ...
2
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1answer
71 views

Asymptote of solution of a differential equation without solving it

Consider the following differential equation (domain $\mathbb{R}$): $$ u(x) = 1 - u'(x) $$ and suppose $u(0) = 0$. How can one prove that $u(x) \to 1$ for $x \to \infty$ without solving the ...
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0answers
16 views

Problem on energy of a Discrete Galerkin Method

I'm reading an article from this website: article question is in page 3,about a wave equation,and use the Galerkin method to discrete the space. (1) page4 why the author use the fraction ...
1
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0answers
14 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
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0answers
21 views

Maximal interval of existence -Exact ODE

If the solution to my ODE is $x(t)=-t+\ln(3/2-t^2/2)$, why is the maximal interval of existence $-(\sqrt 3, \sqrt3)$? The initial conditions are $x(-1)=1$ I have tried many ways of working this out, ...
8
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1answer
58 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
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0answers
15 views

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 ...
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1answer
22 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
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2answers
35 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
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1answer
17 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 ...
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0answers
27 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
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1answer
41 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
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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
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0answers
23 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
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3answers
100 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?
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0answers
24 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 ...
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0answers
16 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
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1answer
54 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 ...
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0answers
25 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 ...
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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
104 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
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1answer
63 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
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1answer
36 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
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4answers
137 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 ...
3
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0answers
60 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
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0answers
21 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: ...
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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
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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: $y=x^{c}\{a_{0}+a_{1}x+a_{2}x^{2}+.....+a_{r}x^{r}+.....\}$ ...
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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
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1answer
22 views

Potential equations with boundary conditions [closed]

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
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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
102 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
32 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 ...