Questions tagged [ordinary-differential-equations]

For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variables. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

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0answers
20 views

Find $(1 − \sin y)H(0, y)$ for a $2$-dimensional autonomous system

I got $\cos^2 y$ for this question; not sure if I got it right. Can someone check it for me? Consider the following $2$-dimensional autonomous system: $$x′ = 3x + \cos y$$ $$y′ = − e^x − 3y.$$ Let $H(...
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0answers
35 views

Find the basis of $V$ where $V$ is the subset of the solutions to a ODE

Let $V$ the subset of the solutions to the differential equation $$\frac{d^{3}x}{dt^{3}}-x =0$$ such that $x(0)=0$. Prove that $V$ is a vectorial space and find a basis for $V$. I found the general ...
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0answers
13 views

Help with Matlab | Ordinary Differential Equations

With the help of Matlab, I want to find the slope field in phase space of the following: a) $\dot{x}=\begin{pmatrix} 2 & -5 \\1 & -2 \end{pmatrix}x$ b) $\dot{x}=\begin{pmatrix} 3 & -2 \\4 &...
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13 views

Qualitative theory of systems of ODE that involve thousands of functions/equations?

I am trying to learn systems biology modelling (https://arxiv.org/pdf/1711.08079.pdf is example article that handle the parameter identification problem and mentions the number in the order of ...
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1answer
16 views

Finding a function that satisfies the ODE: $-2y'_{(x)}=y_{(\frac{1}{x})}$

I would appreciate if someone could elaborate on how I can find the general function (needs to have a second derivative), that satisfies: $$-2y'_{(x)}=y_{(\frac{1}{x})}$$ I can say that the constant ...
2
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1answer
21 views

Verify the solution of first order linear differential equations

The solution of $$y'(t) = a(t)\,y(t)+b(t)$$ is given by $$y(t) = y_0\,e^{A(t)} + e^{A(t)} \int_0^{t}b(s)e^{-A(s)}\,ds$$ I'd like to show it's fulfilling the ode above by the differentiation of it: $$\...
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0answers
25 views

I genuinely do not understand the Newton-Kantorovich method for BVPs :(

I've been struggling to wrap my head around the concept of the Newton-Kantorovich method and the steps necessary to approximate a solution to a BVP: y" = f(x,y,y') on the interval a $\leq$ x $\...
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0answers
15 views

Find the eigenvalues and eigenfunctions of the following boundary value problem.

I have the boundary value problem $$y''(x)= \lambda y(x)$$ $x \in (0,1)$, $\lambda \in \mathbb{R} $ $$y(0)=y'(0)+1$$ $$y(1)=y'(1)+1$$ I want to find the eigenvalues $\lambda $ and the corresponding ...
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0answers
47 views

Solving $ mx'' = -mg - k_1 x' + k_2 (x')^2 $

When trying to solve the tipical problem of free fall of an object with air drag we arrive at the differential equation: $$ mx'' = -mg - k_1 x' + k_2 (x')^2 $$ with initial conditions: $x(0)=0$ and $...
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2answers
45 views

Infinitely many different solutions of differential equation

Give infinitely many different solutions of the differential equation: $y'=3y^2-12xy+12x^2+2$ I'm not really sure where to start with this, I don't really know how I'd got about even finding just the ...
2
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2answers
21 views

Existence and Uniqueness theorem as it applies to finding an explicit solution

If the conditions of the theorem are met for some ordinary differential equation, then we are guaranteed that a solution exists. However, I don't fully understand what it means for a solution to exist....
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1answer
39 views

Trouble solving $y' y''=x^2+x$

I'm stuck on solving $$y' y''=x^2+x$$ with $(x,y,y')=(0,1,2)$. Here's what I've tried $$\int y' y'' \ dx = \int (x^2+x)\ dx$$ by substitution $$\frac{1}{2}\left(\frac{dy}{dx}\right)^2=\frac{x^3}{3}+\...
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0answers
12 views

Using Laplace transforms, solve for x only in the following pair of simultaneous differential equations [closed]

$$ 2·dy/dt− y + x·dx/dt - \sin t = 0 \\ 3·dy/dt+ x - 2·dx/dt - e^t = 0 $$ initial conditions: given that $x(0) = y(0) = 0$
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1answer
28 views

No solution exists on interval for Initial value problem

a) Show that the set of functions S= {$\frac {sin(nx)}n$}$_{n\ge1}$ on $I=[0,1]$ is equicontinuous and uniformly bounded. b) Show that the IVP $y'=f(x,y)$, $y(1)=1$ with $f(x,y)$ = $\begin{cases} x, ...
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0answers
18 views

Linear ODE change of independent Variable [closed]

In solving second order linear ODE using change of independent variable from x to z . We put x=f(z) Then we write this statement, $\frac{dy}{dx} = \frac{dy}{dz}\frac{dz}{dx}$ I tried to rigorously get ...
0
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1answer
27 views

Solving a second-order matrix differential equation - periodic solutions

Let $\frac{d^2}{dt^2}x=\begin{pmatrix}1 &1 \\ 0 &a\end{pmatrix}x$. For which a $\in {\mathbb{R}}$ there exist periodic solutions. I think only for $a<0$ there can be periodic solutions ...
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0answers
13 views

Can the general equation for a simple harmonic oscillator be written as a zero-dimensional general wave equation?

Can you go from this (one-dimensional wave equation): $\frac{d^2y(x,t)}{dt^2} = v^2\frac{d^2y(x,t)}{dx^2}$ to this (general equation for a simple harmonic oscillator): $\frac{d^2y(t)}{dt^2} = -\omega^...
3
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2answers
59 views

What is it to solve an equation forward?

I'm reading a book in Monetary Economics and I don't understand a step. I have this expression: $$ \dfrac{\lambda_{t}}{P_{t}} = \beta \left( \dfrac{\lambda_{t+1} + \mu_{t+1}}{P_{t+1}} \right) $$ And ...
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1answer
14 views

Compartmental models

How can we prove the positivity and the boundlessness of compartments in a compartmental model.
1
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3answers
58 views

Help me find error in ODE for sensitivity analysis of parameters of Lotka-Voltera equation

I have a Lotka-Voltera model on which i want to perform parameter estimation by calculating the gradients of the parameters using an extended ODE system. I know there are different methods for doing ...
0
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1answer
40 views

zeros of Bessel function

Let's denote $J_\alpha$ the Bessel functions of first kind, satisfying the equation $$x^2y''+xy'+(x^2-\alpha^2)y=0$$ Now consider its zeros, there are $2$ questions. For the case $\alpha=0$, find the ...
1
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1answer
23 views

Nonhomogeneous First order differential equation

I'm trying to understand what is wrong with this solution, since I'm not getting the same answer in Matlab $y'-xy=xy^{3/2}\, ,y(1)=4$ \begin{align*} y'-xy=&xy^{3/2}&\\ \dfrac{dy}{dx}=&x(y+...
2
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3answers
63 views

A question about change of variables in an ODE or in general

I have a question about differentiation in this question here: differential equation Cauchy-Euler I understand that it uses product rule to go from the 2nd line to the 3rd line (where the arrow point ...
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1answer
39 views

Eigenvalue of differential equation [closed]

How can I find the eigenvalues to the linear transformation $$T(y) = y''(t) -2y'(t)$$
2
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0answers
40 views

Find the eigenvalues and eigenfunctions of the following integral equation.

I have the integral equation $$u(x)=1+\lambda \int_0^1 K(x,t)u(t)dt $$ $x \in (0,1)$, $\lambda \in \mathbb{R} $ and $$K(x,t)=\begin{cases} x(t+1) & t \leq x \\ t(x+1) & x \leq t \end{cases} $$ ...
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3answers
30 views

Solving $y^{'} + \sqrt{1+{y^{'}}^2}=Ce^{x/k}$

My memories about ODE are rather old. I would appreciate any help driving me to the solution of this equation: $$y^{'} + \sqrt{1+{y^{'}}^2}=Ce^{x/k}$$ where $C$ and $k$ are constants.
-1
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1answer
44 views

runge kutta 2 in python

I am trying to solve an equation in fluid mechanics using the runge-kutta 2 method, usually it seems quite doable but in this case its with x y and z and i cant seem to make the code. Here is what i ...
0
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0answers
18 views

Bound the error of numerical ODE simulation

Let's assume, I have an autonomous ODE of the form $$ \frac{dy}{dt} = \sum \limits_{i=1}^n \alpha_i\cdot f_i(y) $$ which I want to solve numerically in $t\in[0, t_{end}]$ with some initial condition $...
1
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1answer
32 views

Higher order derivatives of Differential function

I am currently working on a presentation about the N-Body problem. I have to solve a specific differential equation through a Taylor Series So the differential equation is: $\begin{align} \frac{d^2}{...
2
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0answers
28 views

Does a restriction to $x$ set when finding $y'$ from solving a second order differential equation affect the domain of the solution? (pursuit curves)

I was solving a problem regarding pursuit curves. The initial differential equation is the following: \begin{equation} xy'' = \sigma \sqrt{1+(y')^2} \end{equation} I used the substitution $v=y'$, thus ...
-8
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0answers
38 views

Solve $𝑦''-4𝑦'-5𝑦=1-2x$

Solve $𝑦''-4𝑦'-5𝑦=1-2x$ Please solve the problem in detail (in steps) about the differential equation
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0answers
17 views

Evolution equations of a 2-gender age-structure model?

This sounds a bit complicated but I want to grab more feelings on age-structured problems. Less than 2 days to the exam so I appreciate any help. So suppose we get 4 age classes in a population census ...
0
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0answers
11 views

The evolution equations under different year classes of a age-structured model?

This sounds a bit complicated but I want to grab more feelings on age-structured problems. Less than 2 days to the exam so I appreciate any help. So suppose we only get 3 year classes in a school at ...
0
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0answers
20 views

Is my solution correct? I tried solving a homogeneoes system of differential equations through MATLAB.

Given the following equations: These equations represent the movement of a particle. The particle starts moving at $t =0$ in $(-2,0)$ at a speed of $(1,2)$. Solving the problem with the following ...
0
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0answers
22 views

A differential equation of second order

Find $y:I \to \mathbb{R}$ for $I \subseteq \mathbb{R}$, such that $$ \begin{cases} y''(t) = \frac{t}{1 + (y(t))^2 + (y'(t))^2} \\ y(0)=1 \\ y'(0)=2 \end{cases}$$ How does one go about solving this? ...
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2answers
38 views

Solving coupled ODEs

Context: This is part of one of the derivation I'm trying to do for turbulent wake flows. The book contains the governing equations and the answer but with my limited mathematical knowledge, I'm not ...
1
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1answer
23 views

About Green function on ODE IVP, BVP

I am sophomore student learning ODE. While learning ODE, suddenly met Green function in IVP, BVP. My 1st question is why it is introduced in IVP, BVP, such as: (Due to reduction of order & ...
1
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1answer
12 views

Existence and uniqueness of IVP solutions, vector-valued equations

The existence and uniqueness of a solution to the IVP $$\underline{y'}=f(x,\underline{y}),\enspace \underline y(0)=\underline{y_0},$$ where $\underline{y} = (y_1, y_2,...,y_n)$, is guaranteed if the ...
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0answers
26 views

Finding $\lim_{t\to\infty}y$ for the initial value problem $2\frac{dy}{dt}-y=4 \sin(3t)$ with $y(0)=y_0$

I have the initial value problem $$2\frac{dy}{dt}-y=4 \sin(3t), \quad y(0)=y_0$$ I have to determine $\lim\limits_{t \to \infty} y$. I found the solution of the homogeneous problem and then a ...
0
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1answer
27 views

Series representation of the Associated Laguerre Polynomials

I've solved the Associated Laguerre Equation through the Power Series Method, $$ xy''+ (α+1-x)y'+ ny $$ and I end up with the following recurrence relation between the cofficients: $$ a_{k+1} = \frac{(...
2
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1answer
32 views

Find the unique solution to the IVP $x' = Ax$ where $A = \begin{bmatrix} {-3}&{2} \\ {-1}&{-1}\end{bmatrix}$

I began this problem by evaulating $x' = Ax$. Let $$ x' = \begin{bmatrix}{x_1} \\ {x_2}\end{bmatrix}.$$ Then we have $$x'_1=-3x_1 + 2x_2, $$ $$x'_2=-x_1 - x_2, $$ $$x_1(0)=1,$$ $$x_2(0)=-2.$$ From the ...
0
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1answer
44 views

Identification of important information to draw phase portrait of a dynamical system [closed]

If we have a planar dynamical system, the following are the minimum number of information needed to draw phase portrait: Equillibrium point, nullcline, direction field or vector field and Manifold....
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0answers
31 views

Need help solving Differential Equation problem. Use Laplace transform to solve the IVP [closed]

Use the Laplace transform to solve the IVP: $$\frac{dy}{dt} - y = 2\cos(5t),\qquad y(0)=0.$$ enter image description here If anyone can help me with this and provide the steps I would really ...
1
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0answers
25 views

Why does this finite difference schema converge even though it should be unstable?

Consider $u''+u=0$ with the boundary conditions $u(0)=1$ and $u(\frac{3\pi}{2})=-1$. One possible finite difference formula is $u(x)+\frac{u(x-h)-2u(x)+u(x+h)}{h^2}$, which gives the matrix $\begin{...
-2
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1answer
32 views

How do I classify the critical points of the system $x'= y+y^2e^x $; $y'=x?$ [closed]

I have found the critical points to be (0,0) and (0,-1) but I am unsure of how to classify the points such as stable, unstable, etc.
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0answers
8 views

How to study the positivity and the boundlessness of compartment in epidemiology

I want references, books, articles ... or anyone who can help to study the positivity and the boundlessness of the compartments in this STIMR model. I want to study the stability also. Any answer ...
2
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1answer
42 views

General solution for this differential equation

I want the general solution for this equation L and I'm stuck. we have this: $$I(y) = y , D^k(y) : = \frac{d^k}{dx^k}(y) = \frac{d^k y}{dx^k} $$ the equation is : $L(y):= \bigl(D-5I\bigr)^4\bigl(D^2+...
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0answers
14 views

Proving the uniformly asymptotically stability of a null solution

I would like to show that the null solution $ψ(t) ≡ 0$ of the equation $\frac{dx}{dt} = − \frac{x}{t}$, $(x,t)∈R×[a,+∞)$, $(a >0)$, is uniformly stable, asymptotically stable, but not uniformly ...
0
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0answers
30 views

Approaching a 2nd order, non-linear ODE with variable coefficients: $f''(x) + (1-B^2x^2)f(x) - f(x)^3 = 0$

Given a differential equation: $$f''(x) + (1-B^2x^2)f(x) - f(x)^3 = 0$$ subject to the boundary conditions $$f(x=0) =0 \qquad f(x\to\infty) = \gamma_A$$ where $\gamma_A$ is some constant between 0 and ...
0
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0answers
15 views

How do I calculate a critical attractor point?

I have to show that there are two critical points in the following model, but I don't know how to prove it in this model, N=S+I+R. The problem statement is: As plausible values for the parameters (...

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