Ode Definition and 1000 Threads

I struggle with a certain computation. Consider a homogeneous linear differential equation with constant (complex) coefficients. Such an equation is associated with a polynomial ##p(t)##, which we can write $$p(t)=(t-c_1)^{n_1} (t-c_2)^{n_2}\cdots (t-c_k)^{n_k},$$where ##n_1,n_2,\ldots,n_k## are...

I am searching for a solution ##R(r)## of the following radial par1 of the Klein-Gordon equation $$[1]\qquad -\left(1-\left(\frac{r_0}{r}\right)^3\right)^{\frac{4}{3}} \partial_r^2 R(r) - \frac{2}{r \left(1-\left(\frac{r_0}{r}\right)^3\right)^{\frac{2}{3}}} \partial_r R(r) +...

When trying to model a neuronal dendrite or axon, the cylindrical shape is preferred and cable core conductor theory is applied. Considering only passive current, that is, non-voltage-dependent current, such as leakage potassium current, the general form of the cable equation is ∂²V_m/∂X² =...

I'm wondering whether a differential equation that can be integrated numerically forwards in time can also be integrated backwards in time starting from the final state and inverting the momenta/velocities? I tried and it didn't work. But I'm not sure whether I'm making a mistake with my solver...

For this problem, Does someone please know how they combined the two equations for H to get the finial equation ##H(x,y) = .... = c##? Thanks!

Please consider the following step in a proof: After transposing the matrix, its coefficients still are functions of ##x##. Why then the solution ##a, a_1, a_2## is constant?

For this problem, I don't understand why they include the constants of integration ##c_1 and c_2##, since the formula that we are meant to be using is ##\vec x = e^{At}c + e^{At}\int_{t_0}^{t} e^{-As} F(s)~ds## so we already have the integration variables. Does anybody please know why? Thanks!

For this problem, I can confused why they don't include the case where ##b = 0## since ##b < 0##. That is, why don't they include ## λ^2e^{λt} + e^{λt} = 0## when solving the associated homogenous equation? This gives the commentary solution ##u_h = \cos t + \sin t## which is not included in...

I have never solved a matrix ODE before, and am wondering if solving it is similar to solving ##y'=ay## where ##a## is a constant and ##y:\mathbb{R} \longrightarrow \mathbb{R}## is a function. The solution is right according to wikipedia, and I am just looking for your inputs. Thanks...

The set of equations is I have first tried to solve only first two equations (removing the components of other 4 equations from them.) This is the output, where first column is the time, 2nd - X_p, and 3rd - X_n

In case of an integral ##\rightarrow## differential equation of the type: $$ f(t) = \int_0^t g(f(\tau)) d\tau $$ $$ \rightarrow \frac{df(t)}{dt} = g(f(t)) $$ which turns out not to be solvable in exact form because ##g(f(t))## is a non-polynomial function (but it would if ##g(f(t))## was a...

Hello! I am trying to solve the time dependent Schrodinger equation for a 2x2 system and I ended up with this ODE: $$y''=-iA\sin{(\omega t)}y'-B^2y$$ with the initial conditions ##y(t=0)=0## and ##y'(t=0)=B##. I can look at it numerically but I was wondering if there is a way to get something...

Solve the linear system of ODE ##x' = 2x + 3y##, ##y' = -3x + y## with initial conditions ##x(0) = 1, y(0) = 2##.

I'm aware that I can introduce the perimeter p = dy/dx then I can rearrange my equation to make y the subject, then I can show that dp/dx = p/x. However, this only gives me a bunch of quadratic curves for my solution. However given part b I see that two curves are meant to intersect each point...

My code is as follows: but when I use the function in my command window exactsol(t) and input a tolerance but there is an error in LINE 19 saying unrecognized ivpfun, could someone help me fix it as I am unsure of how to proceed from here. function y = exactsol(t) y = zeros (2,1); y(1) =...

My approach is to use the definition of the Force with ##\displaystyle F = \frac{dp}{dt} = \dot{m} v + m \dot{v}##. Since ##m(t)## decreases linearly, I should be able to set ##m(t) = M - \Phi t##, thus ##F = - \Phi v + (M - \Phi t) \dot{v}##, which gives ##\displaystyle v = -\frac{ F - (M -...

All right, we got $$ y'' + y = 4x \sin x $$ We are doing the Complexification $$ \tilde{y''} + \tilde{y} = 4x e^{ix} $$ Complementary function: $$ \begin{align*} \textrm{characteristic equation =}\\ m^2 + 1 = 0 \\ m = \pm i \\ \tilde{y_c} = c_1 e^{ix} + c_2 e^{-ix} \\ \end{align*} $$ Q(x)...

The characteristic equation ## m^3 -6m^2 + 12m -8 = 0## has just one single, I mean all three are equal, root ##m=2##. So, one of the particular solution is ##y_1 = e^{2x}##. How can we find the other two? The technique ##y_2 = u(x) e^{2x}## doesn't seem to work, and even if it were to work how...

I am trying to solve this two level (Schrodinger) equation as a function of time:$$i\begin{pmatrix} \dot{x}\\ \dot{y} \end{pmatrix} = \begin{pmatrix} 0 & iW+dE_0sin(\omega t)\\ -iW+dE_0sin(\omega t) & \Delta \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}$$ (I can go into more details about...

Hi all, I have another second order ODE that I need help with simplifying/solving: ##p''(x) - D\frac{e^{\gamma x}}{A-Ae^{\gamma x}}p'(x) - Fp(x) = 0## where ##\gamma,A,F## can all be assumed to be nonzero real numbers and ##D## is a purely nonzero imaginary number. Any help would be appreciated!

Hello, I'm posting here since what follows is not about homework, but constitutes a personal research which underlies some more general questions. As with the infamous "casus irreducibilis" (i.e. finding the real roots of a cubic function sometimes requires intermediate calculations with...

Hello, I am currently planning on self studying math analysis with MIT ocw courses, but I cannot find the analysis course. I have found on the web that analysis and ODE are the same thing. If I want to get a good introduction to analysis, is the MIT ODE course fit for such a use? Thanks

I believe I am doing everything right up until the point where I have to try and find a recurrence relation. I honestly have no idea what to do from there. I've listed my work in getting the powers of n and the indicies to all match. Any help appreciated. Here is the original DE...

I believe it is the case that any linear second order ode with at most 3 regular singular points can be transformed into a hypergeometric function. I am trying to solve the following equation for a(x): where E, m, v, k_{y} are all constants and I believe turning it into hypergeometric form will...

Hey all, I am currently struggling decoupling (or just solving) a system of coupled ODEs. The general form I wish to solve is: a'(x)=f(x)a(x)+i*g(x)b(x) b'(x)=i*h(x)a(x)+j(x)a(x) where the ' indicates a derivative with respect to x, i is just the imaginary i, and f(x), g(x), h(x), and j(x) are...

Hi PF! Given the following ODE $$(p(x)y')' + q(x)y = 0$$ where ##p(x) = 1-x^2## and ##q(x) = 2-1/(1-x^2)## subject to $$y'(a) + \sec(a)\tan(a)y(a) = 0$$ and $$|y(b)| < \infty,$$ where ##a = \sqrt{1-\cos^2\alpha} : \alpha \in (0,\pi)## and ##b = 1##, what is the Green's function? This is the...

Hello! Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...

Ignoring the second part of the question for now, since I think it will be more clear once I understand how this equation is homogeneous. According to my textbook and online resources a first-order ODE is homogeneous when it can be written like so: $$M(x,y) dx + N(x,y) dy = 0$$ and ##M(x,y)##...

I'm studying ODEs and have understood most of the results of the first chapter of my ODE book, this is still bothers me. Suppose $$\begin{cases} f \in \mathcal{C}(\mathbb{R}) \\ \dot{x} = f(x) \\ x(0) = 0 \\ f(0) = 0 \\ \end{cases}. $$ Then, $$ \lim_{\varepsilon \searrow...

Here is my paper. A criticism and other comments are welcome. Abstract: The Lagrange-D'Alembert Principle is one of the fundamental tools of classical mechanics. We generalize this principle to mechanics-like ODE in Banach spaces. As an application we discuss geodesics in infinite dimensional...

I am trying to find a way to prove that a certain first order ode has a unique solution on the interval (1,infinity). Usually the way to do this is to show that if x' = f(t,x) (derivative with respect to t), then f(t,x) and the partial derivative with respect to f are continuous. However, this...

this is pretty easy for me to solve, no doubt on that. My question is on the constant. Alternatively, is it correct to have, ##ln x= \frac {v^2}{2}##+ C, then work it from there... secondly, we are 'making" ##c= ln k##, is it for convenience purposes?, supposing i left the constant as it is...

i am new to MATLAB and and as shown below I have a second order differential equation M*u''+K*u=F(t) where M is the mass matrix and K is the stifness matrix and u is the displacement. and i have to write a code for MATLAB using ODE45 to get a solution for u. there was not so much information on...

so I am trying to solve this equation y'+5.6y=9.5cos(2x)+2.4sin(2x) . I want the c in order to y'(0)=0. I am really lost

I am trying to self study Ordinary Differential Equations and am totally fed up of "cookbook style ODEs". I have recently finished Hubbard's Multivariable Calculus Book and Strang's Linear algebra book. I would like a rigorous and Comprehensive book on ODEs. I have shortlisted a few books below...

Non-homegenous first order ODE so start with an integrating factor ##\mu## $$\mu=\textrm{exp}\left(\int a dt\right)=e^t.$$ Then rewrite the original equation as $$\frac{d}{dt}\mu y = \mu g(t).$$ Using definite integrals and splitting the integration across the two cases, $$\begin{align}...

Sorry the problem is a bit long to read. thank you to anyone who comments. We consider the initial value problem for the Burger's equation with viscosity given by $$\begin{cases} \partial_t u-\partial^2_xu+u\partial_xu=0 & \text{in}\quad (1,T)\times R\\\quad \quad \quad \quad \quad...

I have following differential equation dV/dt = 5 - 2 * V(t)^(1/3) which represents a the time its take to drain a barrel of rain water which contain 25 Liter of water, at t = 0. I am suppose to calculate the least amount of water in barrel during this process. If I set the rate of growth to...

The ODE given to us is y' = xcosy. I am having a bit of trouble when it comes to solving this problem. We are supposed to show that the solution has a local minimum at x = 0 with the hint to think of the first derivative test. However, I am only really familiar with the first derivative test...

I have an ODE, which is y' = 5x3(y-1)1/5 with the initial condition y(0)=1, I must find two solutions. My attempt at solving this problem is as follows: Separate the equation, we get, dy/(y-1)1/5 = 5x3dx. Integrate both sides, ∫ dy/(y-1)1/5 = ∫ 5x3dx. We are left with, (5/4)(y-1)4/5 = (5x4)/4...

I'm new to learning about ODE's and I just want to make sure I am on the right track and understanding everything properly. We have our ODE which is y' = 6x3(y-1)1/6 with y(x0)=y0. I know that existence means that if f is continuous on an open rectangle that contains (x0, y0) then the IVP has...

This is a very simple question: I would like to solve for ##\psi## in this equation $$\frac{d^{2}\psi}{d\xi^2} =\xi^2\psi$$ I so apply ##y=c_{1}e^{-kx}+c_{2}e^{kx}## and ##\psi## should be equal to ##\psi=c_{1}e^{-\xi^2}+c_{2}e^{\xi^2}##, because ##(D^2-\xi^2)\psi=0##. However the answer is...

Hello, I am trying to compute some non-linear equations with pseudospectral/collocation methods. Basically I am expanding the solution as $$ y(x)=\sum_{n=0}^{N-1} a_n T_n(x), $$ Being the basis an Chebyshev polynomial with the mapping x in [0,inf]. Then we put this into a general...

Say you have the set of coupled, non-linear ODEs as derived in this thread, it has two unknowns ##N(t)## and ##\theta(t)##: $$ N - mg = - m\frac{L}{2}\left(\dot{\theta}^2\cos(\theta) + \ddot{\theta}\sin(\theta)\right)$$ $$ \frac{L}{2}N\sin(\theta) = \frac{1}{12}ml^2\ddot{\theta}$$ What freedom...

to I am a bit clueless on how to get break the ##r X(r)## from inside the derivative.P.S. I tried to copy from Symbolab instead of pasting the picture, but it didn't let me.

I'm looking for a general analytical solution to a particular ODE that comes up in neuroscience a lot. My feeling is that such a solution can't be obtained, otherwise someone would have presented it by now, but I don't have a good understanding of why it is so hard to solve. The equations as...

I'm trying to find the local truncation error of the autonomous ODE: fx/ft = f(x). I know that the error is |x(t1) − x1|, but I can't successfully figure out the Taylor expansion to get to the answer, which I believe is O(h^3). Any help would be greatly appreciated!

Actually I was trying to write a small program in Scilab to simulate a quantum particle. When I give a potential higher than energy, the wave function should go like exp(-x) and decay. But my program just increases without bound. Is there any nice way to do anything about it?