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

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Runge-Kutta methods and butcher tableau

What does the Butcher tableau of a Runge-Kutta method tell me about the method, besides the coefficients in its formulation? In particular, what requirements about it guarantee consistency and ...
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Differential equation with $x'(t)=\sqrt[5]{(x(t))}$

Solve the following Cauchy problem for $t\in\mathbb R$ $x'(t)=\sqrt[5]{(x(t))}$ $x(0)=0$ Is the solution unique ? Now this is a differential equation of the form $x'=f(x)$, thus; $\...
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Solution to non-linear OIDE

How do I go about solving this equation? $\frac{\partial F(r,y)}{\partial r} = Q(r,y) - P(r,y) F(r,y) - R(r,y)F(r,y)\int_0^\infty dy'S(r,y') F(r,y')$ with the initial condition that $F(r=0,y) = 0 \ \...
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Explicit solution to a first order nonlinear ODE

Is there any explicit solution to the following ODE? $G'(z) =aG(z)+bG(z)^α-c$ $G(0) = d_0 $
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Asymptotic behaviour of $\varphi''(x)=F(\varphi(x))$

I'm concerned with the discussion of a ODE, especially the discussion of the solution. I've got the assumptions that there is the relation $\varphi''(x)=F(\varphi(x))$ for all $x$ on $\mathbb{R}$. ...
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523 views

Quaternions: Rotation Matrix Derivative

Given Data and Specifications in Question If $q(t)$ represents the position vector as result of rotation with an angular velocity $\omega(t)$ in quaternions, then you can make the relationship $...
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29 views

homogeneous BVP has at most one linearly independent solution

I am trying to understand following proof. I understand the set up however can't make the connection with the Picard Lindelöf Theorem. Can you please help me with this? Statement: The homogeneous ...
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537 views

Differential Equation Find general solution of y'' - y =cosh(x) using variation of parameters

Hello I am having some issues with the simplification of the DE, I am okay up on till $$y_p(x)=v_1(x)y_{p1}(x) + v_2(x)y_{p2}(x) $$ $$ \frac12 e^{-x}\left(\frac {-e^{2x}}4-\frac x2\right)+ \frac 12e^{...
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$M(x,y)\,dx+N(x,y)\,dy=0$ can always be converted to the form $dy/dx=F(y/x)$

I'm being asked to prove that all $M(x,y)\,dx+N(x,y)\,dy=0$ can always be converted to the form $dy/dx=F(y/x)$. My guess is that I just have to manipulate and that's easy. But I have no clue what $F(...
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How do you draw a phase potrait plot when one of the eigenvalues is zero?

So I have the system $X'=\begin{pmatrix}a&1\\2a&2\end{pmatrix}X$ I found the eigenvalues to be $\lambda=(a+2,0)$ and the eigenvectors to be $V=\left(\begin{pmatrix}1\\2\end{pmatrix},\begin{...
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Describe as a system of differential equations

How is it possible to describe the $\dfrac{du_c}{dt}$ and $\dfrac{di}{dt}$ as a system of differential-equations? $$ u_s = u_r + u_l + u_c\\ i = C\frac{du_c}{dt}\\ u_l = L\frac{di}{dt}\\ u_r = Ri\\ $...
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Proving that maximal interval of existence exists and that solution is unque

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ \frac{dy}{dx}=y^{2}+\lambda\sin^{2}x, y(...
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163 views

Qualitative properties of solutions to a ordinary differential equation.

I have this problem : $$\begin{cases} -(p(t)u'(t))'=f(t,u(t))\\u(0)=u(+\infty)=0\end{cases}$$ we have that $u$ is continues, $f:\mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and ...
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65 views

Matrix linearization of the Lagrangian points.

I have to solve a long problem, and I´m in trouble in a step. The step is to linearize the next differential equation, by writtin its correspondient Jacobian, and then, calculate the eigenvalues of ...
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50 views

Ordinary Differential Equation with a trigonometric function: radius of convergence?

For the equation $$x^2y'' + y' + \tan(x)\,y = 0$$ establish lower bounds for the radius of convergence about the point $$x_0 = 1.$$
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Global existence of ode system without solving it explicity.asdf

Here is the ode system that I am looking at $x' = -y-z$ $y' = x + ay$ $z' = b + z(x-c)$ where a,b,c are positive constants. By the local existence theorem, I know that there is a local solution, ...
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83 views

Numerical analysis- Runge Kutta

I have: $$y'(x)= \sin(y); y(0)=1$$ I need to calculate the function values with Runge-Kutta. The range is [0,1]. My problem is that I need to choose the h (=dx) such that the error will be in order ...
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199 views

Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...
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146 views

Using the Lyapunov-Perron method to find the local stable/unstable manifolds

Hello Stack Exchange community. I am currently having an issue finding the local stable/unstable manifolds of this system. After going at it for a few hours I believe the person who wrote this ...
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414 views

Local truncation error of Euler method

Wikipedia and this book say the local truncation error of Euler method is $O(h^2)$. But this book and A friendly Introduction to Numerical Analysis say it's $O(h)$. Which is correct? I have a similar ...
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23 views

Sturm-Liouville equation with rational coefficient

I am trying to solve a regular Sturm-Liouville type Ordinary Differential Equation (ODE) with the following form: \begin{equation} \frac{d}{dy} [q(y) \frac{dW(y)}{dy}]+p(y) W(y) = -\lambda W(y) \end{...
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Show following IVP has solutions.

Show that the solution $y(t)$ of the given initial value problem exists on the specified interval. $$a) y'=y^2+\cos t^2 , y(0)=0; t\in [0,1/2]$$ $$(b) y'=(4y+e^{-t^2})e^{2y},y(0)=0; t\in[0,1/8\sqrt{...
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204 views

stability of FTCS scheme for parabolic equation

Can you suggest any method for stability analysis of FTCS scheme for the the following parabolic equation ? D.E: $u_{t}=a(x,t)u_{xx}+f(x,t,u)$, $0<x<1$, $0<t<T$, $T>0$ BCs: $u_{x}(0,t)...
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Standard notation for Differential Equation Solutions

Quick question: I was solving a differential equation, and wanted to know which of these expressions is in the standard notation for an answer to a differential equation: (a) $y^{3/2} - x^{3/2}=7$ ...
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131 views

Find the fundamental matrix of a system of ODEs?

To linearize a system, in one of the steps I am required to find the fundamental matrix $\Phi$(t) of a system such that $\Phi$(0)=I. The example system my professor used: $\dot{x} = x - y - x^3 - xy^...
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36 views

How t find z (unknown) in Runge-Kutta question

I'm trying to solve the below question solve $\dfrac{dx}{dy}=\dfrac{1}{x+y}$ for $x=0.5$ to $z$ using R-K (order $4$) with $x_0=0$, $y_0=1$ (take $h=0.5$). Please help me and tell me how to ...
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54 views

Blow up solution of a Riccati's equation.

Consider the Cauchy problem $$ \left\{ \begin{array}{l} \dot x=x(t)^2+t\\ x(0)=0 \end{array} \right. $$ Show that its solution is not defined in $[0,3]$.
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Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
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solution of 3rd order non linear differential equation

I need help regarding solution of this equation which has been solved in a research paper but I cant figure it out. Please help $f^{\prime\prime\prime} + 3ff^{\prime\prime}- 3(f^{\prime})^2 - (Ha +\...
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333 views

Finite Difference - Forward Difference with 2nd order Accuracy: What to do at the boundary?

I implemented a BVP using a first-order finite difference scheme after the shooting method did not work reliably. Its the first time I have worked with this. The code works but I would like to move to ...
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I need to model a population where every component have a fixed life span.

I have a certain population, which growth is function of a certain factor. I have already modeled the growth. Now I need to impose that every individual in the population have a fixed life length. Can ...
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Computer Code Friendly Books On Differential Equations?

When I need a differential equation for this or that application I generally search (by hand) through old paper and ink books written by mechanical or electronical (electrical) engineers. Sometimes my ...
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Find the equilibrium of the given ODE

Given the following non homogeneous, autonomous ODE: $$y'(t) = Ay(t)+b $$ with $y(0) = y_0$ where $y_0 \in R^{2m}$ is a known vector. Also , $A$ is $2m$ by $2m$ and invertible, and b $\in R^{2m}$. ...
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Uniqueness of solution for a system of differential equations

A friend of mine working on Auction Theory needs to establish uniqueness of solution (up to initial and boundary conditions) of a system of differential equations of the form $$ F(y_1,y_2,y_3,\dot{y}...
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61 views

Discretization of an Euler-Bernoulli

Given the following Euler-Bernoulli equation: $$ (s(x) w(x)'')''= q(x),\ \ x \in [0,1]$$ Could someone explain why the following discretization scheme may not be a good idea? \begin{align*} (sw'')''...
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Matrix exponent form

We have an equation of matrix exponent $ Ae^{Ax}R-e^{Ax}R (P_1 +P_2 x) = Y \tag1$ Given condition $A,R,P_1,P_2,Y$ are constant $3 \times 3 $ matrices. R is invertible,orthonormal,determinent ...
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Second order, inhomogeneous, linear differential equation

I come across this equation in book $$F(z)=(1-\lambda + \mu )f(z) + (\lambda - \mu) zf'(z) + \lambda\mu z^2f''(z)$$ where $\lambda \not= 0$ and $\mu \not= 0.$ My question is how to find $f$? Can ...
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Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where $X$...
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Uniform perturbative solutions to the Mathieu equation

The Mathieu equation is a second-order linear differential equation given by $$y''(t) + [a - 2q\cos(2t)]y(t) = 0$$ There are two special functions defined as linearly independent solutions to Mathieu'...
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Saari's homographic conjecture and the actual definition of homography

By the wikipedia definitions found here and here, and especially by the definition implicit in this MSE post, it seems two images are homographic if they are renderings of the same set of points in ...
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Help solving particular D.E

I'm going through past exams for revision and couldn't get the same answer as the markscheme for this problem. QP http://papers.xtremepapers.com/CIE/Cambridge%20International%20A%20and%20AS%20Level/...
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Equation of a curve with a local minimum fixed at $x=a$ when we rotate the curve about the origin.

We have a strangely curved plank. If we place a round weighted object on it, it will rest itself at one point of it, when we incline the plank slowly, the object will gradually move towards a resting ...
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Central manifold theorem => Stable/unstable manifold?

I'm a bit confused why we always separate the stable/unstable manifold theorem and the central manifold theorem. The stable/unstable manifold theorem applies to a hyperbolic point ($\mathrm{Re}(\...
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86 views

Verification of Fourier transformation of Io-sinh function

I try to match, but it could not match $I_o-\sinh$ Practical Fourier Transform pair developed by Ben Logan, transform pair also published in The Practical Application of the Fourier Integral ...
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134 views

Integrability of 1-forms and Stokes' Theorem

Let $\alpha$ be a $1$-form defined on a manifold $M$ and $\Delta = ker (\alpha)$. The classical theorem of Frobenius says that $\Delta$ is integrable if $\alpha \wedge d\alpha =0$ i.e if $d\alpha$ is ...
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Stochastic differential equation for a Fokker-Planck-type equation with a non-derivative term

I have something similar to a Fokker-Planck equation of the form $\frac{\partial}{\partial t}f( x,t) = A(x,t)f(x,t)- \frac{\partial}{\partial x}[B(x,t) f(x,t)] +\frac{1}{2}\frac{\partial ^2}{\partial ...
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PDE and Taylor's formula

I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates. Here are the equations: 1) $\frac{\partial^3 f}{\partial x^3}...
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Long-time asymptotic behaviour of a system of two ODEs

We have the following nonlinear ODE: $$ f' = af-bg -(f+g)^k \bigl(f'(0) +g'(0)\bigr) + f'(0), $$ $$ \bigl(G-T(x)\bigr) g' = -af+bg - g'(0), $$ where $a,b,k,G$ are constants, $f'(0)$ and $g'(0)$ are ...
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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 ...
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find matrix A given its exponential

I'm given $e^{At}$ and I need to find A From http://www.math24.net/method-of-matrix-exponential.html I see that $$\frac{d}{dt}(e^{At})=Ae^{At}$$ so does it mean, that to answer my question I just ...