Questions tagged [ordinary-differential-equations]

For questions about ordinary differential equations, which are differential equations containing only derivatives w.r.t. one variable. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

8,898 questions
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On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
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Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to ...
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Application of Thoms transversality theorem

I try to verify example 20.4.10 from Wiggins - Introduction to Applied Nonlinear Dynamical Systems and Chaos and I am quite new to the topic so please be patient. In the book is written that the ...
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Solve ODE $y'' + (y')^2 + y = \ln(x)$

I want to solve $y'' + (y')^2 + y = \ln(x)$ with boundary conditions $y(1) = 0$ and $y(2) = \ln(2)$. The solution is $y = \ln(x)$ but I don't know how to start the problem.
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Solving a dual integral equation involving a zeroth-order Bessel function

Consider the following dual integral equations \begin{align} \int_0^\infty q^3 f_0(q) J_0 (qr) \, \mathrm{d} q &= g(r) \qquad\qquad\quad (0<r<1) , \\ \int_0^\infty f_0(q) J_0 (qr) \, \...
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Recommendation for intro to geometric integrators?

Explicit Request Looking for book or lecture note recommendations on numerical optimization that (ideally) have the following: Emphasis on geometric and physical intuition Emphasis on symplectic ...
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Region of attraction of simple ODE with perturbation

There are a few nice discussions about ROA covering a few subtopics: Region of attraction of : $x'=-y-x^3,y'=x-y^3$ via Lyapunov Function Region of attraction and stability via liapunov&#...
Alan Turing in his original paper presents the following system of differential equations: $\frac{\delta X_r}{\delta t} = f(X_r, Y_r) + \mu(X_{r+1} -2X_r + X_{r-1})$ $\frac{\delta Y_r}{\delta t} = g(... 0answers 84 views How far has a chasing wasp flown as her target walks around a square? I take a walk each morning along the sides of a square; each side is one mile. I start at one corner and walk at a constant speed. As I start on the walk, an unfriendly wasp always starts at the ... 0answers 147 views Asymptotics for solutions of a version of Lienard's differential equation Consider the second order differential equation$ x'' + f(x)x' + g(x) = 0 $with $$f(x) = -\lambda + x^2, \quad g(x) = (-1 + x^2)x \, .$$ with$\lambda > 0$. Note: The original post had a ... 0answers 260 views Two Matlab ODE solvers, two different results I am solving a system of ODEs using Matlab. One particular set of parameters caused the solver to fail, so I worked my way through the different solvers Matlab provides. I was surprised to find that ... 0answers 214 views Numerically solving a non-linear PDE by an ODE on the Fourier coefficients I need to solve numerically a PDE of the form $$u_t(x,t)=u_{xx}(x,t)+u_x(x,t)^2-a(x)u_x(x,t)-a_x(x)$$ with initial condition$u(x,0)=u_0(x)$. I can assume that both$u(\cdot,t)$and$a(\cdot)$are ... 0answers 111 views Question about a (relatively simple looking) differential operator and its eigenvalues A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form$L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$for ... 0answers 95 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: $$\frac{\mathrm{d}^{2}\vec{x}}{\mathrm{d}t^{... 0answers 499 views Proof Strategy for a Dynamical System of Points on the Plane I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ... 0answers 275 views Hints/Help studying an Abel Differential Equation I want to know more than qualitative information about the Abel differential equation \frac{dy}{dx}+y^3+x=0. \qquad ... \;(1) Since I don´t know how to solve this and as far as could see, this ... 0answers 148 views \Sigma-equivalence between (y,z,1) and (y+\mathcal{O}(2),z+\mathcal{O}(2),1) Consider the sets \mathfrak{X}(\mathbb{R}^3) = \{X: \mathbb{R}^3 \to \mathbb{R}^3; X \mbox{ is smooth}\} and \Sigma = \{0\}\times\mathbb{R}^2. Let X, Y be vector fields in \mathfrak{X}(\mathbb{... 0answers 85 views Analytic or perturbative solution in any limits? Consider the system of 3 ordinary differential equations$$\dot{x}=v\dot{v}=a\dot{a}=-Aa+v^{2}-x$$which can also be written as a single 3rd order ODE$$\dddot{x}=-A\ddot{x}+\dot{x}^{2}-... 0answers 71 views Non-unique solution of first order PDE Question: $$\frac{\partial u}{\partial x} \frac{\partial u}{\partial y}=1 \qquad \qquad u=0 \; \text{ when } \; x+y=1$$ Find all possible solutions and state where each one exists. Attempt: Using ... 0answers 44 views Identifying ODE types for solving by hand and when to use computers instead So this questions relates to my specific ODE but also ODEs in general. I am a big fan of solving ODEs by hand, but I also know when to give up and use, say, Mathematica to solve it for me. Having ... 0answers 245 views Prove that doesn't exist a Lyapunov function Given the following ODE in polar coordinates \begin{array}{lcl} \frac{dr}{dt} = r\sin(\frac{1}{r}) \\ \frac{d\theta}{dt} = 1\end{array} 1) Show that the origin$(0,0)$is Lyapunov stable ... 0answers 335 views Useful reformulation of Goldbach's conjecture? Let us assume there exists some infinite order differential equation whose solution is: $$y= \sum_{n=1}^\infty A_n \exp(p_n^sx)$$ Where$p_n$is the$n$'th prime. Substituting$ y=\exp(\lambda x)$... 0answers 78 views Solving linear nonhomogenous system of differential equations I have a question, concerning one special solution of the following system. First I will sum up my results, to give a context for my question. My question will be seperated by a line, if you want to ... 0answers 92 views Generalization of Liouville's formula to other coefficients of the characteristic polynomial If$X(t)$is an$n \times n$matrix solving linear homogeneous ODE $$\frac{d}{dt} X(t) = A(t)X(t),$$ then for$\det X(t)$we have Liouville's formula: $$\frac{d}{dt} \det X(t) = \text{tr} A(t) \det ... 0answers 143 views Linearization which is a Sturm-Liouville problem: Stability questions Consider the scalar phase equation$$ \theta_t=\theta_{xx}+f(\theta),\qquad f(\theta+2\pi).\qquad (1) $$Traveling waves profiles \theta(x-ct) can be found using phase-plane analysis for$$ \theta_{\... 0answers 93 views Closed form solution to an ordinary differential equaiton How to solve the following ordinary differential equation? $$y'(x)= \frac{C_1}{y(x)} +C_2 C_3 \cos\left(C_3 x\right) +C_4$$ where$C_1, C_2, C_3, C_4\in \mathbb{R}$are all constants. It looks ... 0answers 168 views Transforming the solutions of$\dot x = f(\mathbf{x}, t)$and$\dot x = B(\mathbf{x}, t)f(\mathbf{x}, t)$into each other How can I prove the following theorem? If the function$B(\mathbf{x}, t)$is strictly positive, then the solutions of the two differential equations$\dot x = f(\mathbf{x}, t)$and$\dot x = B(\...
In Spain's book, Functions of mathematical physics he introduces the contour integral method of solving ODEs. The baseic idea is: given an ODE $\sum_0^m a_r(t) \frac {d^rf}{dt^r} = 0$, a solution may ...