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

learn more… | top users | synonyms (1)

3
votes
0answers
37 views

the exact graph of the general solution for $x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$

i need someone to give me exact graph of the general solution for $$x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$$ i solved it manually , the general solution is like this ...
3
votes
0answers
63 views

Existence of Periodic Solution

I'm working with the system of equations below that represents a Pendulum with constant forcing. \begin{align*} \theta'&=v\\ v'&=-bv-\sin(\theta)+k \end{align*} Where $\theta$ gives the ...
3
votes
0answers
72 views

How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $ M ^ n $ be a Riemannian manifold, $ f: M \rightarrow M $ be ...
3
votes
0answers
83 views

How can one derive Stokes lines of the Stokes phenomenon of asymptotics from a differential equation?

Is there a standard technique to calculate Stokes lines and anti-Stokes lines of the Stokes phenomenon of asymptotics for a function defined as the general solution to a differential equation without ...
3
votes
0answers
33 views

What allows us to break up $dy/dx$ in solving a separable differential equation?

Suppose you had a separable first order differential equation that can be written as $$ \frac{dy}{dx}=f(x,y)=g(x)h(y) $$ Rigourously, what allows us to rearrange this as $$ \frac{1}{h(y)}dy=g(x)dx? ...
3
votes
0answers
31 views

Second order DE question

I am looking for tips for this equation: $ 4xy''+y'+xy'+\frac{3}{2}y=0 $. I am solving with the substitution y=a(x)b(x), but it is getting messy..
3
votes
0answers
49 views

Finding the general solution to system of linear equations: $y' = 2y,y''=4y-y'$

Question: I want to find the general solution to the following system: $\begin{pmatrix} \dot{y}_1 \\ \dot{y}_2 \end{pmatrix} = \begin{pmatrix} 2&0\\4&-1 ...
3
votes
0answers
46 views

General Solution of DE?

I've got the following ODE, and I'm just having trouble coming up with the form of the general solution. I'm really trying to find the particular solution, but in order to do that, I need to know the ...
3
votes
0answers
48 views

Find a series solution to $(x^2-2)y''+6xy'+4y=0$.

Find a series solution to $(x^2-2)y''+6xy'+4y=0$. A. Find the recurrence relation to $a_n$: My answer is $a_{n+2}=a_n\cdot \frac{n+4}{2(n+2)}$ which is correct. B. Using A, write two independent ...
3
votes
0answers
23 views

Differential Equations: Say whether the equation has bounded solutions at $x = 0$

Its been forever since I've done Diff EQ and I can't remember how to go about solving this problem: Say whether the equation has bounded solutions at $x = 0$ and whether all solutions are bounded: $$ ...
3
votes
0answers
61 views

Existence theorem on weak solutions of ordinary differential equations

Consider an ordinary vector-valued differential equation of the form $$ \begin{align*} \dot y(t) &= f(t,y(t)), \\ y(0) &= y_0 \in \mathbb{R}^n. \end{align*} $$ It is well known that if $f$ is ...
3
votes
0answers
37 views

Solving a system of first order differential equations

So, I have (another) problem with differential equations (from an optimal control problem). I am trying to solve the following system of DEs (is this even a system?): $$ \lambda'(t) = r \lambda(t) + ...
3
votes
0answers
42 views

Let $\eta (x)=\int_0^\infty e^{at}\xi(\phi_t(x)) dt$ then $\eta$ is a $C^1$ function

Consider the following problem. Suppose that $a>0, r >0$ and $\xi:\mathbb R \to [o,\infty)$ is a $C^2$ which vanishes in the complement of the interval $(-r,r)$. Also suppose that ...
3
votes
0answers
47 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} = ...
3
votes
0answers
81 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 ...
3
votes
0answers
99 views

Non-Linear Ordinary Differential Equation in Fluid Dynamics

So while trying to model the physics of a rocket shot from the ground through the atmosphere, I came up with a second-order Non-Linear ODE of the form: $$ \ddot y + \dot y^2 e^y = f(t) $$ This is ...
3
votes
0answers
102 views

Solving 1D telegrapher's equation by reduction to two-dimensional wave equation

The solution $w : \mathbb R \times \mathbb R_{+} \to \mathbb R$ of the Cauchy problem for the telegrapher's equation $$ w_{tt} - c^2 w_{xx} + c^2 \lambda^2 w = 0 $$ with $$ w(x,0) = 0, \qquad ...
3
votes
0answers
64 views

Finding the exact solution of a differential equation

Let $y=f(x)$. Is it possible to find an exact solution of the following differential equation?: \begin{equation} \ddot y+2\dot y-5xy=e^{-2x}\nonumber \end{equation} Many thanks in advance, -- Cesar ...
3
votes
0answers
38 views

Green's function the way George Green defined it

This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x-x')$ function as their source on the RHS. But ...
3
votes
0answers
60 views

Door mechanism differential equation

I have been wondering about a door mechanism I have seen. It has a wire attached to the upper corner of the door and from there to the corresponding corner in the door frame, where a weight hangs from ...
3
votes
0answers
64 views

How do I solve this differential equation?

$y^{(7)}+4y^{(6)}+8y^{(5)}+9y^{(4)}+8y^{(3)}+8y^{(2)}+8y^{(1)}+4y=e^{-x} (5sinx-cosx) $ The characteristic equation $ \lambda ^7 +4\lambda ^6+8\lambda ^5+ 9\lambda ^4 +8\lambda ^3+8\lambda ...
3
votes
0answers
25 views

Lipschitz function in ODE

Let $f:\mathbb{R} \times \mathbb{R^n} \rightarrow \mathbb{R^n}$ lipschitz function. Show given that $(t_0,x_0)\in \mathbb{R} \times \mathbb{R^n} $ there exists a unique solution: $$x' = f(t,x)$$ ...
3
votes
0answers
118 views

Fundamental Solution of a Nonlinear ODE (using Riccati Transformation & Wronskian)

I am given the differential equation: \begin{equation*} y^{\prime}(t) = y(t)^{2} + 2\sin(t)\cos(t) - \sin^{4}(t) \end{equation*} and one solution $y_{1}(t) = \sin^{2}(t)$. I wish to find a second ...
3
votes
0answers
99 views

Cauchy-Euler Equation of order $n$

What I wish to prove is that for a Cauchy-Euler equation of order $n$, the substitution $x=e^{t}$ transforms it into a linear differential equation with constant coefficients. To put it as a theorem: ...
3
votes
0answers
83 views

Differential Equation has a unique solution periodic

Let $A(t)$ continuous and periodic of period $S$ in $\mathbb{R}$. Suppose $x' = Ax$ has $\varphi \equiv 0$ as the only periodic solution of period $S$. Show that there exists $\delta> 0$ such that ...
3
votes
0answers
27 views

Invertible e converges series.

If T is a linear transformation on $R^n$ with $||T - I||<1$, prove that $T$ is invertible and that the series $\sum_{k=0}^\infty(I-T)^k$ converges absolutely to $T^{-1}.$ (Use the geometric series) ...
3
votes
0answers
99 views

WKB and asymptotic behavior of second order differential equation

I want to study the large $x$ solution to a Riccati equation. After listening to the lectures on Mathematical Physics by Carl Bender, I have fallen in love with asymptotic analysis. But, by no means ...
3
votes
0answers
144 views

Are any solutions lost when solving non-exact differential equations?

I have just started studying differential equations, one of the problems I found while I was practicing is "Consider the equation $$ (y^2 + 2xy)dx - x^2 dy=0 $$ (a) Show that this equation is not ...
3
votes
0answers
93 views

Solve ODE by Fourier transform, and versus by Laplace transform?

Regarding solving ODE by Fourier transform, I read a nice reply by O.L.. After applying Fourier transform to an ODE to obtain an algebraic equation, the reply showed that some terms involving the ...
3
votes
0answers
51 views

Show that this initial-value problem has a unique solution

I am trying to show that the following initial-value problem $$\frac{dx}{dt} = - x + tx^{1/2}; \quad x(2) = 2$$ has a unique solution on $I = [2,3]$. By letting $f(t,x) = - x + tx^{1/2}$ and $(t_0 ...
3
votes
0answers
71 views

How to integrate/differentiate parameters in differential equations

I have an ODE, which I would be fine about solving, were it not for the parameter: $$(\omega^2+x^2)\frac{dy}{dx}=y$$ I'm given that $\omega>0$ is a parameter. Separating the variables gives: ...
3
votes
0answers
46 views

Would the transformation of a differential equation obey the same algebra?

I've found that the algebra of this differential equation $$\frac{d^2y}{dz^2}-(3z^2+\gamma)\frac{dy}{dz}+(cz+\alpha)y=0$$ is in $sl(2)$ because it is possible to use the generators of the $sl(2)$ ...
3
votes
0answers
84 views

Measurability of points regular

I'm reviewing the proof of the theorem of oseledet the book Mañe: Let $M$ a compact metric space and $f:M \rightarrow M$ a homeomorphism, $\pi: F \rightarrow M$ a finite-dimensional continuos vector ...
3
votes
0answers
106 views

Does the implicit function theorem imply Peano existence theorem

In The implicit function theorem written by Krantz & Parks, it's said that the implicit function theorem implies the following existence theorem of ODE: Theorem 4.1.1 If $F(t,x)$, ...
3
votes
0answers
48 views

Qualitative dependence of solution to second-order matrix differential equation on eigenvalues

Suppose we have a matrix differential equation in $\vec{x}(t)=\left(\begin{smallmatrix}x_{1}(t) \\ \vdots \\ x_{n}(t)\end{smallmatrix}\right)$, such that: ...
3
votes
0answers
125 views

ODE with delta function

Consider the following ODE $$y''+a\delta (x)y+\lambda y=0$$ subject to the initial conditions $$y(\pm\pi )=0$$ (1) Show that there is a set of eigenvalues $$\tan (\pi \sqrt{\lambda ...
3
votes
0answers
52 views

Merton's Problem Stochastic Differential Equation

Solve the following numerical case of Merton's optimal portfolio selection problem: find an optimal policy function $(s, y) \mapsto u(s, y)$ such that for the Ito diusion determined by $dX_t =X_t(u(t, ...
3
votes
0answers
76 views

solve nonlinear second order ODE

I obtained Nonlinear second order differential equation as $y\cdot y''+y'^2-m\cdot y^{-a}y'^2+k=0$, Where $y'= \dfrac{dy}{dx}$, $y''=\dfrac{d^2y}{dx^2}$. I could not obtain the solution so please ...
3
votes
0answers
139 views

Lyapunov Function and $\omega$-limit sets

I will ask you about a particular equation but what I would really enjoy is an (if possible, comprehensive) answer to the following question : How can we, using a Lyapunov function, study the ...
3
votes
0answers
57 views

dropping a particle into a vector field, part 3

Okay, so I've been independently trying to study basic systems of differential equations as they relate to dropping a particle into a vector field. I have had two previous posts on the matter trying ...
3
votes
0answers
195 views

A probable inspiring proof to Poincare lemma

Poincare lemma says if a smooth $p$-form $\omega$ is closed, then $\omega$ must be exact. Let's put it in another way, it says the solution of $d\omega=0$ is $\omega=d\eta$ for some $(p-1)$-form ...
3
votes
0answers
78 views

Singular solution to $(x+2y)y'=1$

I have a problem and I got most of the solution, but don't understand how to proceed. The problem is to solve: $$(x+2y)y'=1, \qquad y(0)=-1.$$ Here is my reasoning: Substitute $z = x+2y$. Then ...
3
votes
0answers
212 views

Causality in Dirac delta forced harmonic oscillator

If I take the simple forced harmonic oscillator equation, apply the Fourier transform to both sides, and assuming the forcing function is a Dirac delta function (at the origin) I get: $ F(s) = \frac ...
3
votes
0answers
44 views

Perturbation Theory on Finite Domains

In this video (from 27.00 - 50.00, which you don't need to watch!) a guy shows how you can solve the general second order ode $y'' + P(x)y = 0$ using perturbation theory. However he points out that ...
3
votes
0answers
76 views

Sturm-liouville problem, first eigenvalue

Any idea to solve the Sturm-Liouville Problem $$ -\cos^{2}(t)g''+n\sin(t)\cos(t)g'-(n+1)\cos^{2}(t)g=(\delta)g, $$ with $t\in[\epsilon,0]$, and boundary conditions $g(\epsilon)=g(0)=0$? We may ...
3
votes
0answers
114 views

A photon in expanding Universe (a snail on a tree)

I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
3
votes
0answers
100 views

Differential equation $y'(t) = 1-y(t) e^{y(t)-1}$

I am interested in finding a clean explicit solution (if possible) to the differential equation $$ y'(t) = 1-y(t) e^{y(t)-1}, $$ where $0 \le t < 1$ and $0 \le y \le 1$. This can obviously be ...
3
votes
0answers
82 views

System of many non-linear (quadratic) first order O.D.E. (numerical strategy or simplification)

I have a large system (N>100) of equations $\frac{d\vec{P}}{dt}= A(t) + B(t) \vec{P} + \vec{P}^T C(t) \vec{P}$ where $\vec{P}$ is a vector of N functions of the variable t. What is the correct ...
3
votes
0answers
472 views

Solve a differential equation using Fourier series

Assume I have a second order differential equation $\ddot{x} = F(x,\dot{x})$ (or an equivalent equation of first order) and that I know there is a periodic solution to it (for simplicity's sake, ...
3
votes
0answers
457 views

George Simmons' “Differential Equations with Applications and Historical Notes” vs. “Differential Equations: Theory, Technique, and Practice”

I've heard much acclaim for George F. Simmons' "Differential Equations with Applications and Historical Notes" (2nd edition). I've noticed there's a newer book by Simmons and Krantz entitled ...