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

learn more… | top users | synonyms (1)

2
votes
0answers
56 views

Constructing a function using the Fourier transform

Pick an integer $n\ge 5$ and let $f\in C_{C}^{\infty}(\mathbb{R}^{N})$. We want to use the Fourier transform to formally construct a function $u\in L^{\infty}(R^{n})$ that solves $\Delta^{2}u(x)=f(x)$...
2
votes
0answers
45 views

Leading behaviour of DE at infinity

This is taken from the book of Bender and Orszag, problem 3.44. Find the leading behavior as $x\rightarrow+\infty$ of the differential equation: $x^3y'' - (2x^3 -x^2)y' +(x^3-x^2-1)y=0$ Explain ...
2
votes
0answers
49 views

What problems are related with the following type of FDE with delay?

Consider the following class of functional differential equations with delay: $$\begin{align} \frac{du}{dt} &= F(x,t,u(x,t),u_{t,x}), & (x,t) &\in [a,b] \times [0,T] \\ u(x,t) &= \psi(...
2
votes
0answers
49 views

Why does an infinite Neumann boundary condition become a Dirichlet condition?

Often when I read a paper I see a statement of the type: Our boundary condition at the surface is $\frac{\partial f}{\partial x} = \alpha$. In the limit of $\alpha \to \infty$ this is equivalent ...
2
votes
0answers
50 views

Writing a 2nd order linear ODE as a set of 2 first order ODEs in its independent solutions

Given the homogeneous linear ODE $$y''(x)+ P(x) y'(x) + Q(x) y(x)=0$$ where $x\in(0,\infty)$ and $P$ and $Q$ are some smooth (but not necessarily bounded) functions. I know that we can write this as a ...
2
votes
0answers
137 views

How many solutions does Riemann's P-symbol describe?

The Papperitz-Riemann P-symbol $$ \tag 1 y(z) = P \left\{ \begin{matrix} z_1 & z_2 & z_3 & \; \\ \alpha_1 & \alpha_2 & \alpha_3 & z \\ \beta_1 & \beta_2 & \...
2
votes
0answers
45 views

Existence and uniqueness of solution of the ODE

Consider the initial value problem $(1)\left\{\begin{matrix} y'(t)=y^2 &, 0 \leq t \leq 2 \\ y(0)=1 & \end{matrix}\right.$. Verify that the following theorem: "Let $c>0$ and $f \in C([a,...
2
votes
0answers
43 views

A solution of the ODE $\theta''(t)+\sin(\theta)=0$ with $\theta(0)=0$ is an odd function

Consider the equation $$\theta''(t)+\sin(\theta)=0, \theta(0)=0,\theta'(0)=\alpha>0$$ Prove that $\theta(t)=-\theta(-t)$ for every $t\in \mathbb{R}$. It's a homework problem. I have shown ...
2
votes
0answers
130 views

Solution of non-linear Fredholm(Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question but got no response. I rephrase it so that anyone who knows operator theory and integral equations would help me out.....I faced a problem in physics which is a non-linear ...
2
votes
0answers
75 views

What should I know before reading Ordinary Differential Equations of VI Arnold?

I have just adquired some math books and VI Arnold is one of them. But it was more sophisticated that I may think, so I want to know what should I know to be able to read it. Actually I have on my ...
2
votes
0answers
29 views

Translated Laplace transform

Is there any way to rewrite the Laplace transform is such a way that that one can apply to an IVP not centred at zero, that is, at some $y^{(n)}(a_n) = b_n$ for $n\in\mathbb{N}$ and $a_n \in\mathbb{R}\...
2
votes
0answers
38 views

Is there a fiber bundle approach to nonlinear oscillations?

I've recently been learning about nonlinear oscillations, and I noticed a seemingly strong connection between how the equations of motion are solved/approximated, and fiber bundles (or vector bundles ...
2
votes
0answers
56 views

Are there any symplectic integration techniques that are A-stable (work on stiff equations)?

The first and second Dahlquist Barriers show that (paraphrasing): Explicit multi-step methods cannot be A-stable and thus are not accurate for stiff equations. Implicit multi-step methods will only ...
2
votes
0answers
55 views

Estimating limit cycle of ODE system

I'm looking at a system of ODEs: $$\dot{x} = -y - \epsilon^2 x + xy^2$$ $$\dot{y} = x -\epsilon^2 y - x^2$$ After plotting these in Matlab I can see there is a limit cycle very close to the origin ...
2
votes
0answers
58 views

Solving first order non linear ODE

I am trying to solve the following first order non-linear differential equation: $$ \frac{\partial y}{\partial x} = -\sqrt{\frac{2(\sigma + 1)}{\sigma}} \sqrt{-\frac{1}{2y^{2}} + \frac{1}{8y^{8}}+\...
2
votes
0answers
51 views

How to solve this non-linear, second order ODE

does anyone know how to solve this ODE? $ yy'' +y' +y =0 $ where y is a function of one real variable.
2
votes
0answers
32 views

sufficient conditions for finite time of existence of integral curves of a vector field

Let $U\subset \mathbb{R}^2$ open, $\partial U\neq \varnothing$, $V\colon U\rightarrow \mathbb{R}^2$ smooth. Let $c\colon [0,t_{max})\rightarrow U$ be an integral curve of $V$, where $t_{max}$ is the ...
2
votes
0answers
77 views

How to solve this initial-boundary value problem for a PDE

Consider $$u_{tt}-a^2u_{xx}+u_t+a u_x=0,\quad 0<x<\infty,\quad t>0,(*)$$ where $u_t=\frac{\partial u}{\partial t}$ and etc. It is not so hard to use the method of characteristics to solve it ...
2
votes
0answers
58 views

How to solve the ODE $2x\frac{dy}{dx}=C(1+(\frac{dy}{dx})^2)^2$?

I am struggling with this ODE I obtained when solving the Euler-Lagrange equation. Can any one help me with solving the ODE $$2x\frac{dy}{dx}=C(1+(\frac{dy}{dx})^2)^2$$ Thanks so much! It comes ...
2
votes
0answers
45 views

Time taken to empty a hemispherical shaped tank

The tank has a radius of $2$m when initially filled and has an outlet of cross section $12$ cm2 Outlet flow as I calculated goes according to the law $V(t)=0.6\sqrt{2gh(t)}$. Having found out the ...
2
votes
0answers
212 views

How to solve this recurrence relation and solving the power series

Take the following recurrence relation into account: $$ a_{n+2} = \frac{1}{(n+1)(n+2)} \sum_{k=0}^n (s_k - (k+1)a_{k+1})(n-k+1)a_{n-k+1} $$ I know that: $$ s_{2m+1} = \frac{(-1)^m}{(2m+1)!} $$ and ...
2
votes
0answers
44 views

How to solve $\int_{x}^{x+a} f_X(u) du=e^{-2\lambda_1 x} \int_{x-a}^{x} f_X(u) du$

How to solve equation of the type \begin{align*} \int_{x}^{x+a} f(u) du=e^{-\lambda x} \int_{x-a}^{x} f(u) du \end{align*} we want to solve for $f(x)$ where $\lambda,a$ are some constants. Things I ...
2
votes
0answers
23 views
2
votes
0answers
35 views

How to find first-order quasi-linear PDEs form second-order quasi-linear PDE?

Transform $u_{tt} u_{xx}-u^{2}_{tx} + uu_{tt} + 1=0 $ into first-order quasi-linear PDEs. Attempt: $u_{tt}(u_{xx}+u)=(u_{tx}-1)(u_{tx}+1)$ To get $u_{tt} = u_{tx}-1\Rightarrow u_t = u_x +F(t)$, ...
2
votes
0answers
367 views

Differential Equation Direction field

What i want to achieve: I want to plot the direction fields of the following three differential equations: 1. Malthusian growth model: $p'(t)=\lambda*p(t)$ with $\lambda=1$ and $p(t)=t$ 2. Linear ...
2
votes
0answers
19 views

Convergance of DASPK for a non-linear DAE

I have a system of non-linear DAE and I noticed that the system does not converge if some of the equations are not differentiated. For example, if the control volume equation is represented as this: $...
2
votes
0answers
34 views

What is the general way to rewrite an ODE with respect to a change in coordinates?

I have an ODE : $y' = f(x, y)$ I change for coordinates $(r, s) = (g(x, y), h(x, y))$. What is the equation like in terms of r and s ? If it can help, in my case, $(r, s) = (y.x^{-k}, ln(x))$. I can ...
2
votes
0answers
8 views

Differential Equations of Transformed System

At the moment I'm struggling with a problem I found in a script to one of my lectures: Let $\phi \in C^\infty(\mathbb{R}^{2n})$ have the property that the system $p_i=\frac{\partial}{\partial q_i}\...
2
votes
0answers
66 views

A line integral equation popped up when trying to derive Exact ODE integrating factor, can it be solved analytically?

(For convenience, for any functions, only its first instance the x,y dependence will be written out, all subsequent instance the x,y will be suppressed) I have an ODE $$M(x,y)+N(x,y)\frac{dy}{dx}=0$$ ...
2
votes
0answers
57 views

book suggestion on manifolds

I've to learn differential equations on Manifolds. Can any one please suggest some books/lecture notes for differential equations on Manifolds ?
2
votes
0answers
45 views

Stability of non-autonomous stochastic differential equation

I'm looking for a good reference or insight to under what conditions can I prove stability (or instability) for the following general n-dimensional non-autonomous stochastic differential equation: $...
2
votes
0answers
34 views

Determining linear independence of three simple functions for a third order ODE. (2.9-7)

This is a very similar post to one previous by me but I felt that not all questions were satisfactorily answered. But I am sincerely grateful to those who tried. I would like a sharp independent eye ...
2
votes
0answers
52 views

suggestion for lyapunov function

Consider differential equation \begin{align}x'&=-t(x+y)\\ y'&=-y+x-y(y^2-6).\end{align} Can some one suggest a lyapunov function for it. I have examined $V(x,y)=x^2+y^2$ , $V(x,y)=(1+t^2)(x^...
2
votes
0answers
144 views

Find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ $$y(0)=...
2
votes
0answers
41 views

Is this a valid equivalence between the classes of Differential Equations?

Consider the general first order Linear Ordinary Differential Equation: $$ \frac{dy}{dx} = A(x,y) = \frac{F(x,y)}{G(x,y)}$$ This equation is characteristic equation of the Partial Differential ...
2
votes
0answers
45 views

Two linear independent functions have zero Wronski

Suppose $f$ and $g$ are linear independent $C^1$ functions on $[a,b]$ and there Wronski det is zero, i.e. $$fg^{'}-f^{'}g=0$$ Can we say there exist an point $t_0\in [a,b]$ such that: $$f(t_0)=f^{'}...
2
votes
0answers
23 views

Why is the basin of an asymptotically stable solution to a differential equation an open set?

Why is the basin of an asymptotically stable solution to a differential equation an open set? The basin is defined as the set off all points s.t. their limit at $t = \infty$ is equal to some solution $...
2
votes
0answers
78 views

Pursuit Curve, Parametric Equation

So its a classic problem: Object $A$ starts at the origin $(0,0)$ and moves straight up the $y$ axis with a speed $v$. Object $B$ starts at point $(1,0)$, always moves towards object $A$ and has a ...
2
votes
0answers
83 views

a system with differential equations

I have a system which is described by the following differential equation. I want a closed form formulas to calculate $v_1(t)$ and $v_2(t)$ with the given parameters. In the following equations $p, k, ...
2
votes
0answers
61 views

Recursive differential equations

Suppose we took the odd solution to $y''+y=0$ which is $\sin(x)$. If we put this in place of $y$ in the differential equation we get the equation: $$y''+\sin(y)=0$$ the odd solution to which is an ...
2
votes
0answers
152 views

best book for practicing unsolved problems in differential equation and linear algebra

I started reading differential equation and linear algebra. Can anyone provide the link/book name where I may get many questions to practice. Generally, in the end of book only few problems are there. ...
2
votes
0answers
63 views

Equilibrium points and linear stability

Consider the nondimensional amplitude equation for $A = A(t)$ where $t$ is time given by (1): $$ \frac{dA}{dt} = \sigma A - a_1 A^3 - a_3 A^5 = f(A) \text{ with } \sigma \in \mathbb{R}, a_1 < 0, \...
2
votes
0answers
51 views

Why does this nonlinear ODE solution not work?

I am relatively new to Python and trying to use it to solve a second order nonlinear differential equation, specifically the Poisson-Boltzmann equation in an electrolyte. $$\phi''(r) + \frac2 r\phi'(...
2
votes
0answers
289 views

Identifying Hamiltonian Systems with Phase Portrait

the following is a homework question (that isn't going to be graded) and I'm not sure how to do it. I know that the solution trajectories cannot cross the H(x,y)=constant curves, but I'm not sure ...
2
votes
0answers
40 views

BVP eigenvalue problem

I am working on the following problem and I am completely stuck: Show that the eigenvalue problem $$ -u''+4\pi^{2}\int_{0}^{1} u(x)\,dx=\lambda\,u $$ with $u(0)=u(1)$ and $u'(0)=u'(1)$ has $\lambda=4\...
2
votes
0answers
33 views

Stability of an equilibrium solution with 0 denominator

I'm testing the equilibria of a differential equation and found that one has a 0 denominator. Example: $$\frac{dx}{dt}=2x^{(1/2)}-5$$ Which, when you try and evaluate the derivative at 0, you end up ...
2
votes
0answers
133 views

Stuck trying to solve wave equation in $n$-dimensions.

Solving the wave equation $u_{tt} = c^{2} \Delta{u}$ subject to $u(0,x) = f(x)$ and $u_{t}(0,x) = g(x)$ gives us d'Alembert's formula. I'm looking to solve the wave equation, subject to these same ...
2
votes
0answers
57 views

solving this differential equation for $y$, Is it even possible?

Lets say I have the following: \begin{gather} \frac{(y')^3 + 3 y' y'' + y'''}{(y')^2 + y''} = \sqrt{1+(y')^2}\\ \frac{((y')^3+3y' y'' + y''')^2}{((y')^2 + y'')^2} = 1+(y')^2\\ \frac{(y')^6 + 6 (y')^4 ...
2
votes
0answers
36 views

Initial conditions to solve an ODE?

Given is the following inhomogenous linear ODE (4th order): $$q_0\cdot\sigma + q_1\cdot \dot\sigma + q_2\cdot \ddot\sigma + q_3\cdot \dddot\sigma + q_4\cdot\ddddot\sigma = p_0\cdot\epsilon + p_1\cdot\...
2
votes
0answers
45 views

Solving $2y'''(t)+3t\ y(t)=0$.

For a certain problem, I am trying to solve the ODE $$2y'''(t)+3t\ y(t)=0$$ I am pretty clueless what to do here, any hint would be appreciated. Thank you very much.