Ode Definition and 1000 Threads

Question: So I got around on doing this example, and I'm pretty sure I messed up somewhere, would appreciate if someone could point out what I did wrongly. 1) For any second ODE, I should let: ##y_{1}= y ## and ##y_{2}= y' ## Hence, ##y_{1}'= y' = y_{2} ## and ##y_{2}'= y'' = xy(x)+x^2-y(x) =...

With the new variable, I got: $$p^2 (p'_y)^{2}=k^2(1+p^2)$$ where ##p'_y## is ##\frac{dp}{dy}##. I modified the equation so the variable p and dp can be separated from dy. Here what I got: $$\frac{p}{\sqrt{p^2+1}} dp=k dy$$ I substitute ##p^2+1=u## so I got $$\sqrt{u}=ky+c_1$$ Back substitution...

This equation, is non linear, non-separable, and weird. I would like to have a direction to start working on this. I tried writing sin(2y) = 2sin(y)*cos(y). See, ##xy' = x^3sin^2(y)-2sin(y)cos(y)## Can't separate. Writing in this way: ##(x^3sin^2y-sin2y)dx-xdy=0## Also, I checked that it is...

Suppose I'm solving $$y''(t) = x''(t)$$ where $$x(t)$$ is the ramp function. Then, by taking the Laplace transform of both sides, I need to know $x'(0)$ which is discontinuous. What is the appropriate technique to use here?

Greetings, I have a question to the following section of the book https://www.springer.com/gp/book/9783319163741: I understand that the equation is separable, since I can just write $$ \int_{x_0}^{x} \frac {1}{V(x', \xi, \eta)}dx' =\int_{0}^{t}dt' .$$ However, without knowing the exact shape...

Homework Statement:: ODE -> Transfer Function Assistance Relevant Equations:: Newtonian physics, buoyancy, drag [Mentor Note -- thread moved to DE from the schoolwork forums, since it is for work and not schoolwork] Hello all, I'm new here but I'm looking for a bit of guidance with a...

Hello all, this question really has me and some friends stomped so advice would be appreciated. Ok so, the relevant (dimensionless) continuity equation I have found to be $$\frac{\partial\rho}{\partial t} + (1-2\rho)\frac{\partial \rho}{\partial x} = \begin{cases} \beta, \hspace{3mm} x < 0 \\...

\[ \dfrac{dy}{dx} =\dfrac{x^2+3y^2}{2xy} =\dfrac{x^2}{2xy}+\dfrac{3y^2}{2xy} =\dfrac{x}{2y}+\dfrac{3y}{2x}\] ok not sure if this is the best first steip,,,, if so then do a $u=\dfrac{x}{y}$ ?

I OK going to do #31 if others new OPs I went over the examples but? well we can't 6seem to start by a simple separation I think direction fields can be derived with desmos

Hi I need to solve an equation of the form $$\dot{X}(t) = FX(t) + X(t)F^T + B$$ All of these are matrices. I have an initial condition X(0)=X_0. However, I have no idea how to proceed. How can I make any progress?

let ##y= \sum_{k=-∞}^\infty a_kz^{k+c}## ##y'=\sum_{k=-∞}^\infty (k+c)a_kz^{k+c-1}## ##y"=\sum_{k=-∞}^\infty (k+c)(k+c-1)a_kz^{k+c-2}## therefore, ##y"+y'\frac {1}{z}+y[\frac {z^2-n^2}{z^2}]=0## =##[\sum_{k=-∞}^\infty [(k+c)^2-n^2)]a_k + a_k-2]z^{k+c} ## it follows that...

Hi again, The previous problem was done using y′′(t)+2y′(t)+10y(t)=10 with with intial condition y(0⁻)=0. In the following case, I'm using an initial condition and setting the right hand side equal to zero. Find y(t) for the following differential equation with intial condition y(0⁻)=4...

Find the only periodic solution for 𝑦′+𝑦=𝑏(𝑥) with 𝑏:ℝ→ℝ has a period of 2𝑇 and is 1 for 𝑥(0,𝑇) and −1 for 𝑥(−𝑇,0). The ODE is easy to solve: 𝑦(𝑥)=exp(−𝑥)⋅𝑐+1 and 𝑦(𝑥)=exp(−𝑥)⋅𝑐−1. But how can I find the 𝑐 such that the solution is periodic with a period of 2𝑇? The solution is...

$$y''(x)+y'(x)+F(x)=0$$ Pleas me a idea

I fell upon such an equation : $$-E'(v)a(1+\frac{cE(v)}{\sqrt{E(v)^2-1}})=vE(v)+c\sqrt{E(v)^2-1}$$ It's not separable in E on one side and v expression on the other. So I'm looking for methods to solve this maybe changes of coordinates ?

I have an ODE: (x-1)y'' + (3x-1)y' + y = 0 I need to find the solution about x=0. Since this is an ordinary point, I can use the regular power series solution. Let y = ## \sum_{r=0}^\infty a_r x^r ## after finding the derivatives and putting in the ODE, I have: ## \sum_{r=0}^\infty a_r...

I've been solving these two ODEs ##\frac{d}{d\,r}\,A=F(A,r) + \epsilon f(r)## and ##\frac{d}{d\,r}\,A=F(A,r)##. If the solutions are respectively ##A_1(r,\epsilon)## and ##A_2(r)## then will ##A_1(r,0) = A_2(r)## ? I realize the answer could depend on the actual functions but with the ones...

From Griffiths E&M 4th edition. He went over solving a PDE using separation of variables. It got to this ODE \frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)= -l(l+1)\sin \theta \Theta Griffths states that this ODE has the solution \Theta = P_l(\cos\theta) Where $$P_l =...

I'm fine with this up to a certain point, but I'm not certain if I'm using the substitution correctly. After finding the homogeneous solution do I plug in x= e^t in the original equation and then divide by e^2t to put it in standard form before applying variation of parameters so f=1, or do I...

Hi PF The following ODE $$\ddot x + x - x^3 = 0\\ x(0)=0,\,\,\,\dot x(0) = \frac {1}{ \sqrt 2}$$ is solve exactly with ##\tanh (t/\sqrt 2)##. However, when I try to solve this with MATLAB ode45 (ode23t looks identical) or Mathematica NDSolve I get an oscillatory numerical solution (see...

The problem of interest at the moment is the solution of a simple damped oscillator problem, xddot+2*zeta*wn*xdot+wn^2*x = 0 Everything seems to work fine provided I include the values of zeta and wn inside the function that defines the derivative values. But zeta and wn are actually calculated...

I'm new to using Octave 5.1.0, and a bit confused about how to solve ODEs with Octave. Let me show you a bit of code that I grabbed off a university web site: >> function xdot = pend(x,t) % pend.m xdot(1) = x(2); xdot(2) = - x(1) - 0.1*x(2); end >> sol=lsode( "pend",[0.1, 0.2], t =...

Hi PF! Attached are two plots: signal.pdf is a solution from the Duffing ODE, and plots vertical displacement over time, both the raw signal (blue) and the reconstructed signal from an FFT (red). I've also shown a zoomed in view so you can see how oscillatory the signal is. pow.pdf plots the...

I was recently working on a problem of Griffiths and in the solution's manual it used an argument to solve a diffential equation that caught my attention. It said that it would look first to the steady state solution of the ODE. I tought "All right, I get that" but when I got to translate the...

When reading through Shankar's Principles of Quantum Mechanics, I came across this ODE \psi''-y^2\psi=0 solved in the limit where y tends to infinity. I have tried separating variables and attempted to use an integrating factor to solve this in the general case before taking the limit, but...

I want to solve $\d{y}{x}=\frac{3*(2x-7y)+6}{2*(2x-7y)-3}.$ I don't know its step by step solution. But using some trick of solving ordinary differential equation (which I saw on the Internet), I got the following solution:- $-\frac{17}{21}*(3x-2y)+ln(119y-34x-48)=C$. Now how to solve this...

I have to solve the equation above. I haven't heard about an exact method so I tried to apply perturbation theory. I don't know much about it so I would like to ask for some help. First I put an ##\epsilon## in the coefficient of the non-linear ##\xi^2(t)## term: ##\ddot{\xi}(t)=-b\xi...

The ODE is: \begin{equation} (y'(x)^2 - z'(x)^2) + 2m^2( y(x)^2 - z(x)^2) = 0 \end{equation} Where y(x) and z(x) are real unknown functions of x, m is a constant. I believe there are complex solutions, as well as the trivial case z(x) = y(x) = 0 , but I cannot find any real solutions. Are...

2000 Convert the differential equation $$\displaystyle y^{\prime\prime} + 5y^\prime + 6y =0$$ ok I presume this means to find a general solution so $$\lambda^2+5\lambda+6=(\lambda+3)(\lambda+2)=0$$ then the roots are $$-3,-2$$ thus solutions $$e^{-3x},e^{-2x}$$ ok I think the Wronskain...

Mathematica gives this solution but how does it calculate it? What's the method here?

dM/dY = x+2y+1 dN/dx = 1 (My-Nx)/n = 1 Integrating Factor => e^∫1dx= e^x (xye^x+ye^x+ye^x)dx + (xe^x+2ye^x)dy = 0 dM/dY =xye^x+e^x+2ye^x dN/dx = xye^x+e^x+2ye^x Exact ∫dF/dy * dy = ∫ (xe^x+2ye^x)dy F = xy*e^x + y^2*e^x + c(x) dF/dx = xy*e^x + y*e^x + y^2 * e^x + c'(x)...

I am attempting to solve an ODE using a Calculus add-in for Excel. I am an industry professional and I have not even thought about Differential Equations in 8 years. The equation that I am attempting to solve is in the form: (1) The ODE solver that I am using solves equations of the form...

given the differential equation $\quad y''+5y'+6y=0$ (a)convert into a system of first order (homogeneous) differential equation (b)solve the system. ok just look at an example the first step would be $\quad u=y'$ then $\quad u'+5u+6=0$ so far perhaps?

I have written some ODE solvers, using a method which may not be well known to many. This is my attempt to explain my implementation of the method as simply as possible, but I would appreciate review and corrections. At various points the text mentions Taylor Series recurrences, which I only...

Summary: Looking for guidance on how to model an RLC circuit with a system of ODES, where the variables are the resistor and inductor voltages. This is a maths problem I have to complete for homework. The problem is trying to prove that the attached circuit diagram can be modeled using the...

This is another application of using Taylor recurrences (open access) to solve ODEs to arbitrarily high order (e.g. 10th order in the example invocation). It illustrates use of trigonometric recurrences, rather than the product recurrences in my earlier Lorenz ODE posts. Enjoy! #!/usr/bin/env...

I want to solve ##\frac{du^2}{d\theta ^2}+u=\frac{GM}{h^2}## for ##u(\theta)##, where ##\frac{GM}{h^2}=constant##. The given equation is a nonhomogeneous second order linear DE. I begin by solving the associated homogeneous DE with constant coefficients: ##\frac{du^2}{d\theta ^2}+u=0## which...

For Initial Value problems I want to implement an ODE solver for implicit Euler method with adaptive time step and use step doubling to estimate error. I have found some reading stuff about adaptive time step and error estimation using step doubling but those are mostly related to RK methods. I...

How can an ODE be symmetric? How would you plot an ODE to show off this property? (i.e. what would be the axes?)

According to the wiki entry 'Kuramoto Model', if we consider the ##N=2## case then the governing equations are $$\frac{d \theta_1}{dt} = \omega_i + \frac{K}{2}\sin(\theta_2 - \theta_1)~~~\text{and}~~~\frac{d \theta_2}{dt} = \omega_i + \frac{K}{2}\sin(\theta_1 - \theta_2),$$ where ##\theta_i##...

As you can see, I've tried using KCL at node A to find the 2nd order ODE that describes this circuit in terms of the capacitor voltage. The problem I run into, however, is that I can't find anything to put the node voltage at A in terms of. I've tried (not shown here) doing mesh current as well...

Hi PF! Given the ODE system ##x'(t) = A(t) x(t)## where ##x## is a vector and ##A## a square matrix periodic, so that ##A(t) = A(T+t)##, would the following be a good way to solve the system's stability: fix ##t^*##. Then $$ \int \frac{1}{x} \, dx = \int A(t^*) \, dt \implies\\ x(t) =...

I'm trying to plot the solution to an ODE (with given initial values) but there are some constants in it that I want to evaluate with sliders and I'm not sure what is the right syntax for this. Manipulate[Plot[solution1[t], {t, 0, 10}, PlotRange -> {-Pi, Pi}], {{a, 1, "Driving amplitude"}, 0...

I'm trying to solve the following nonlinear second order ODE where ##a## and ##b## are constants: $$\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}-\frac{y}{ay+b}=0$$ It looks somewhat like the modified Bessel equation, except the third term on the left makes it nonlinear. I've been trying to...

Mentor note: Moved from non-homework forum to here hence no template. So I was able to solve part 1.A of the first problem by hand, the phase portrait is a sideways parabola. However, I want to also show on this on mathematica. I want to solve the equation first and then plot the phase...

Homework Statement how do we solve the ode ## y'+y^2=-2, y(0)=0## using adomian decomposition method?Homework EquationsThe Attempt at a Solution ##Ly = -2-y^2## ## y= 0 + L^{-1}[-2-y^2]## ##y_{0}= -2t## ##y_{1}= -L^{-1}[4t^2] = -4t^3/3## are my steps correct so far in trying to get the Adomian...

I am trying to solve the following first order ODE using a simple Fortran code : $$ ds/dt=k_i * \sqrt{v}$$ where both (ki) and (v) are variables depending on (h) as follows $$ k_i=\sqrt{χ/h^2}$$ $$v= \mu h$$ where (μ) and (χ) are constants. (the arbitrary values of each of them can be seen...

Salutations, I have a problem when I approach this ODE: $$\left(\frac{y}{y'}\right)^2+y^2=b^2\left(x-\frac{y}{y'}\right)^2$$ I have done a series of steps as I show in this link: https://drive.google.com/file/d/1Ht4xxUlm7vXqg4S5-wirKwm7vTESU3mU/view?usp=sharing But I'm not convinced that those...

Hi all, I am a bit new in this, am trying to learn DE, dynamical systems, & chaos. I am looking into some answers for the following questions: 1) Is it always possible to derive a difference equation for every differential equation, and if so how do we do that? 2) Consider Lorenz system...

Hi PF! Given the ODE $$f'' = -\lambda f : f(0)=f(1)=0$$ we know ##f_n = \sin (n\pi x), \lambda_n = (n\pi)^2##. Estimating eigenvalues via Rayleigh quotient implies $$\lambda_n \leq R_n \equiv -\frac{(\phi''_n,\phi_n)}{(\phi_n,\phi_n)}$$ where ##\phi_n## are the trial functions. Does the...