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

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26 views

Find the points at which an IVP admits at least two solutions

Given the IVP: $$\frac{dy}{dx} = x + |\sin(y)|$$ $$y(x_0) = y_0$$ Find the points in $\mathbb R^2$ at which this IVP admits at least two solutions. Clearly, $f(x,y) = x + |\sin(y)|$ is Lipschitz ...
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80 views

Solving second order nonhomogeneous differential equation with non-constant coefficients using Laplace Transform

$ty''(t) + y'(t) -ty(t)= tf(t)$ How to solve the problem using Laplace Transform? Using Laplace transform I got $$Y(s)= C(s^2-a^2)^{-1/2} + (s^2-a^2)^{-1/2}\int (s^2-a^2)^{-1/2}F(s)\,ds$$ where ...
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62 views

How can one prove the existence and uniqueness of solutions to linear differential equations?

It is a theorem (I think) that the equation: $$\mathbf{x}'(t) = A(t)\mathbf{x}(t) + \mathbf{b}(t); \qquad \qquad \mathbf{x}(t_0) = \mathbf{x}_0$$ Has a unique global solution for any matrix ...
2
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153 views

System of first order ODEs with coherent sinusoidal time varying coefficient

I have encountered equations of the form $$\frac{{d{\bf{y}}(t)}}{{dt}} = \left( {{A_0} + {A_1}\cos (\omega t)} \right){\bf{y}}(t)$$where ${\bf{y}}$ is a vector and ${{A_0}}$ and ${{A_1}}$ are square ...
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47 views

Differential Equation $(1+x^2)y'-2xy=\cot(x)$ after integrating factor

$$(1+x^2)y'-2xy=\cot(x)$$ or $$y'=\frac{2x}{1+x^2}y+\frac{\cot(x)}{1+x^2}$$ if I use an integrating factor $(e^{\int\frac{-2x}{1+x^2}dx}=\frac{1}{1+x^2})$ I get $$\frac{y}{(1+x^2)}=\int\frac{\cot ...
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36 views

Solving differential equation with small parameter

I am trying to solve the following equation $$\frac{x}{x+1}\frac{d^{2}\left(\phi^2\right)}{dx^{2}}+\frac{2x+1}{(x+1)^{2}}\frac{d\left(\phi^2\right)}{dx}=\frac{1}{3\phi}$$ with $x\ll1$ and ...
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19 views

Is one dimesional cubic NLS is globally wellposed in $H^{s}(\mathbb R), (0<s<1)$?

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = ...
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30 views

Zeroes of a solution to a differential equation

Show that any solution to the equation $y''+xy=0$ has at least 15 zeroes on the interval $[-25,25]$. Please give me a hint.
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21 views

Is the following complex value differential equation always has a solution?

Let $a(z)$ be a fixed complex value complex variable function, not necessarily holomorphic. Consider the following differential equation $$ \frac{\overline{\partial}f}{\partial \overline{z}}+af=0. $$ ...
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17 views

Linear ordinary differential equations and their evolution operators for measurable operators

Consider the following homogeneous IVP: $$\begin{cases} \dot{u}(t)+A(t)u(t)=0 \\ u(0)=u_0 \end{cases}$$ for $u:[0,1]\to \mathbb{R}^n$ (some interval to some finite dimensional Hilbert space, let's ...
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37 views

Differential equation of the form to find

Lets $f(z)$ is some analytic function on complex plane and $y(z)$ is known analytic function on complex plane. Problem statement: find all $f(z)$ that: $$f(z) = f(z\frac{\partial}{\partial z})y(z)$$ ...
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29 views

Proving the Bessel function solves the Bessel equation

Using the notation for the Bessel function as $J_n(z)=\sum \limits_{k=0}^{\infty}\frac{(-1)^kz^{n+2k}}{k!(n+k)!2^{n+2k}}$, I want to show that $w=J_n(z)$ satisfies ...
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81 views

Periodic Solution of Damped Pendulum with Constant Torque

I have a system of ordinary differential equations $ \theta' = v$ $ v' = -bv - \sin \theta + k$ These are the equations for a pendulum with $\theta$ being angular position, and $v$ being angular ...
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84 views

Eigenvalues for $y''+2y'=\lambda y$

I must find the eigenvalues and eigenfunction for $$y''+2y'=\lambda y$$ with initial conditions $y(0)=0$, $y'(1)=0$. I have found the non-trivial case, and made an attempt to solve for $\lambda$, but ...
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36 views

Eigenvalue Function of Laplace Equation discretizes by nine-point stencil

I'm trying to plot the eigenvalue function of the Laplace equation $$-u_{xx}-u_{yy}=0,\;(x,y)\in (0,1)^2$$ with $$u(x,y)=0$$ on the boundary of the unit square. I have the nine-point stencil ...
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56 views

Using Green's Function to Find Particular Solution

We have the non-homogeneous differential equation $x^3y'''-3x^2y''+6xy'-6y=4x^2$ with conditions $y(1)=1, y'(1)=1, y''(1)=0$, and I have been tasked with finding its particular solution using Green's ...
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70 views

Show the IVP has a unique solution

Assume that $f:\mathbb{R}^{n}\times \mathbb{R} \to \mathbb{R}$ satisfies (i) there exists a constant $M>0$ such that $|f(x,t)-f(y,t)|\leq M|x-y|$ for each $x,y\in \mathbb{R}^n$ and each $t\in ...
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48 views

For time varying linear ODEs: is there a transformation which can make the system admit an exponential solution?

First, I'll start with the properties of the matrix in question: Assume we are given some matrix $\mathbf{A}(t)$, which is time dependent. This matrix is square and not invertible. A system: $$ ...
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46 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 ...
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16 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 ...
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42 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) &= ...
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36 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 ...
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30 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 ...
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54 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 & ...
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35 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 ...
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86 views

Existence and uniqueness of strong solution of stochastic differential equation.

I am currently going through the proof of the existence of a solution of the SDE \begin{align} dX_t = bdt + \sigma dB_t \end{align} where $B_t$ is a Brownian motion wrt a filtration ...
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32 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 ...
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50 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 ...
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25 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 ...
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24 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 ...
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19 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 ...
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41 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 ...
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51 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}} + ...
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44 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.
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25 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 ...
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41 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 ...
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54 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 ...
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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 ...
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139 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 ...
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42 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 ...
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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 ...
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92 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 ...
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11 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: ...
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25 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 ...
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7 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 ...
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42 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
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0answers
27 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
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0answers
29 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
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0answers
38 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 $V(x,y)=x^2+y^2$ , ...