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

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How can I prove that the DE $y'=y^\alpha$ has infinitely many solutions?

I need to show that the DE $y'=y^{\alpha}$, where $\alpha$ is a constant with $0<\alpha<1$, has infinitely many solutions passing through the point $(0,0)$. Also I need four of such solutions. ...
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2answers
667 views

Generalized “Worm on the rubber band ” problem

I found this « Worm on the rubber band » problem in Concrete Mathematics book. A slow worm $W$ starts at one end of a meter-long rubber band and crawls one centimetre per minute toward the other end. ...
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1answer
21 views

general conditions for reverse poincare inequality

I'd like to know when the reverse Poincare inequality is true: Given a bounded domain $\Omega$, and $f \in L^2(\Omega)$, under what conditions on $f$ (neglecting the trivial constant case) and/or ...
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0answers
14 views

IVP Using Numerical Methods

Suppose that $y(t)$ is the exact solution of the ivp $$y'(t)=f(t,y(t)), y(0)=y_0$$ and $u(t)$ is any approximation to $y(t)$ with $u(0)=y(0)$. Define the error $e(t)=y(t)-u(t)$. How can I show ...
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2answers
32 views

Differentiaion Calculus: Trig Inverse Function.

Well, yesterday at a Mathematics exam, i had to find $\frac{dy}{dx}$ of a cotangent inverse function $$ y=\text{arccot}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-sin x} ...
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1answer
83 views

Proof of existence and uniqueness of the exponential function using ODEs

In our lecture notes for our complex analysis class, we were given the following theorem: Theorem: There exists a unique complex function $f$ such that $f(z)$ is a single valued function ...
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1answer
64 views

Proving ODE is periodic if and only if p(t) is periodic

Consider the first-order nonautonomous equation $x′ = p(t)x$ where $p(t)$ is differentiable and periodic with period $T$. Prove that all solutions of this equation are periodic with period $T$ if and ...
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2answers
84 views

find an approximate solution, up to the order of epsilon

The question is to find an approximate solution, up to the order of epsilon of following problem. $$y'' + y+\epsilon y^3 = 0$$ $$y(0) = a$$ $$y'(0) = 0$$ I tried to solve the given problem using ...
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1answer
110 views

$y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$

Let , $y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$ , where, $$f(x)=\begin{cases}1 & \text{ if } 0\le x\le 1\\0 & \text{ if } ...
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2answers
86 views

Inverse Laplace transform of s/s-1

Finding the inverse laplace transform: $$L^{-1}\left\{\frac{s}{s-1}\right\}$$ I wrote: $$L^{-1}\left\{\frac{s}{s-1}\right\}=L^{-1}\left\{\frac{1}{s-1}\right\} + L^{-1}\{1\}=L^{-1}\{1\} + e^{t}$$ And ...
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59 views

Seemingly easy Ordinary Differential Equation

For which values of $T$ can we find a unique solution of the ODE $x''(t) = −x(t) $ satisfying the boundary conditions $x(0) = a_1$ and $x(T) = a_2$ for any values of $a_1$ and $a_2$ ? I can solve ...
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1answer
265 views

How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
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2answers
112 views

Numerical Approximation of Differential Equations with Midpoint Method

I want to proof that the local truncation error of the Midpoint Method is $d_{k+1}=O\left(h^{3}\right)$ Approach The local truncation error is defined as: ...
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1answer
70 views

Modelling population with $\frac{dP}{dt}=P(\beta - \delta P)$

The population $P(t)$ of a biological species can be modelled by $$ \frac{dP}{dt}=P(\beta - \delta P) $$ subject to $P(0)=P_0$ where $\beta$ is the birth rate and $\delta$ is the death rate. ...
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2answers
49 views

Differentiation - simple case

In the book calculus made easy, page 22 the case of the negative power for $y=x^{-2}$ $$\begin{align} y+dy & =(x+dx)^{-2}\tag{1}\\ \\ & = x^{-2}\left(1+\frac{dx}{x}\right)^{-2}\tag{2} ...
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3answers
344 views

Tricky Separable Differential Equation

Please guide me: $y' + ay +b = 0$ (a not zero) is supposed to be separable and has solution $y = ce^{-ax} - \frac ba$ Here is my start to this problem: $\frac{dy}{dx} + ay = -b$ is as far as I can ...
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2answers
378 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } ...
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2answers
133 views

How to solve this differential equation system?

The following system is given: $$ \dot{x} = y + z \\ \dot{y} = x + z \\ \dot{z} = x + y $$ The first thing I did was to find out the eigenvalues. I found out, that -1 is a doubled and 2 a single ...
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1answer
107 views

Solving differential equations using power series

I need to solve this differential equation by power series: $$y''+3xy'+(2x^{2}+6)y=0$$ Any help is great! Thanks!
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173 views

Differential equation with random variable

How can I derive analytically or compute numerically the solution to following differential equation $$ dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X $$ where X is a random variable (e.g. from a ...
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3answers
248 views

I.V.P $y'=\sin(e^{y}), y(0)=a$

Is the I.V.P: $$\begin{cases} \dfrac{dy}{dx}=\sin(e^{y})\\[8pt] y(0)=a \end{cases} \text{ where } a\in \mathbb{R}$$ a) Find the values ​​of $a$ for which $y(x, a)=0$ b) Prove that if $a=0$ then ...
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1answer
209 views

First order non linear Ordinary differential equations

Consider the first order differential equation $\displaystyle\frac{dy}{dt} = f(t,y)= -16t^{3}y^{2}$, with the inital condition $y(0)=1$ Estimate the lipschitz derivative for the differential ...
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2answers
244 views

how did he conclude that?integral

So the question is : Find all continuous functions such that $\displaystyle \int_{0}^{x} f(t) \, dt= ((f(x)^2)+C$. Now in the solution, it starts with this, clearly $f^2$ is differentiable at every ...
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10k views

Solving differential equation $x^2y''-xy'+y=0, x>0$ with non-constant coefficients using characteristic equation?

Whenever you deal with non-constant coefficients you usually use Laplace transform to solve a given differential equation, at least that's how how I learned it. But how would you solve the equation ...
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3answers
553 views

Is $f'(x)-3f(x) = 0$ subspace of differentiable functions $f\colon (0,1)\to \mathbb{R}$

$V$ is space of differentiable functions $f(0,1) \to \mathbb{R}$ and $W$ is a subset of $f$ that meets $f'(x) - 3f(x) = 0$ for all $x\in (0,1).$ Is subset $W$ a subspace of $V$? I know that I have ...
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1answer
295 views

Multistep ODE Solvers

Write both a fourth order Adams Bashforth and Adams Moulton procedure to solve $$x'(t) = x(t)-y(t)-\exp(t);$$ $$y'(t) = x(t)+y(t)+2\exp(t)$$ with initial values $x(0) = -1, y(0) =- 1$ on the ...
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2answers
341 views

Second Order Differential Equations e times sin particular solution

The differential equation I am trying to solve is $$ \dfrac{d^2y}{dt^2} + 4\dfrac{dy}{dt} + 20y = e^{-2t}\sin(4t) $$ I know how to start off. I have done the $s^2 + 4s + 20 = 0$ to get $s = -2-4i$ ...
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1answer
85 views

Solutions and attraction regions of following odes?

Assume a mapping $X: \mathbb{R} \to \mathbb{R}^d$. We know that the solution to ode $$ d X_t = (\mu - X_t) dt $$ is $X_t = (X_0-\mu) e^{- t} + \mu$, which indicates that $X_t$ converges to $\mu$ as ...
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3answers
105 views

Solve $y'-\int_0^xy(t)dt=2$

I have not idea how to approach this differential equation. $$y'-\int_0^xy(t)dt=2$$. Basically, I did, $$F''(t)-F(x)+F(0)=2 \;\;\;\;\;\;\; F'=y$$ I am stuck. Thank You.
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3answers
447 views

General Solution for a given system of equations

Find the general solution of this system of equations: $$x' = \pmatrix{-1&0&0\\1&0&-1\\1&1&0}x$$ I got the eigenvalues to be: $\lambda = -1,\pm i$ The eigenvectors ...
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2answers
684 views

Using Octave to solve systems of two non-linear ODEs

How to solve following system of ordinary differential equations using Octave? $$\frac{dx}{dt} = - [x(t)]^2 - x(t)y(t)$$ $$\frac{dy}{dt} = - [y(t)]^2 - x(t)y(t)$$ Update: initial conditions: ...
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2answers
5k views

Difference between improper node and proper node for phase portrait

Can someone offer an explanation for the difference between these two? I see pictures of what seem to be examples of both, but it's hard for me to discern what a new portrait would be. Any help? ...
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1answer
310 views

Cubic population model question

Have the following population model, $$ \frac {dN}{dt}= cN(N-k)(1-N)$$ The first stage of the question is to investigate the steady states, however im a little stuck on finding the solution to the ...
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43 views

Differential equations - Hypergeometric function [duplicate]

I would like to solve the equation $(x^2-x-6)y''+(5+3x)y'+y=0$ at $x=3$. I think we have to solve this problem in considering the Gauss's hypergeometric equation on the form (*) ...
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0answers
80 views

How do I show that $y(t)=\int_0^tv(t-s)f(s) ds$ is a solution of $L[y]=f(t)$?

We have the $n$th-order scalar differential equation $$L[y]=\frac{d^{n}y}{dt^{n}}+a_1\frac{d^{n-1}y}{dt^{n-1}}+\dots+a_ny=f(t).$$ Let $v(t)$ be the solution of $L[y]=0$ which satisfies the initial ...
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1answer
40 views

how to construct a (2nd order) ODE that will be satisfied by a provided fundamental set?

If given a couple of functions and asked to construct an ODE of the form $y'' + q(x)y' + r(x)y = 0$ admitting of that couple as a fundamental set, once we've established that the couple could be a ...
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3answers
41 views

Converting a differential equation

Consider an ODE $\frac{dy}{dx}=h(x,y)$ such that $h(rx,ry)=h(x,y)$, which implies $h(x,y)=k\left(\frac{x}{y}\right)$. (Why?) Show that this ODE can be changed to a separable ODE for $u=u(x)$, ...
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3answers
308 views

Differential Equations Skydiver Problem

I've seen many variants of this problem online, but not quite the same as this, so I don't believe this is a duplicate. The famous differential equation problem models a skydiver jumping out of a ...
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50 views

How to solve this first-order differential equation?

I have been trying to solve a differential equation as a practice question for my test, but I am just unable to get the correct answer. Please have a look at the D.E: $dy/dx = 1/(3x+\sin(3y))$ My ...
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2answers
68 views

trouble with non-homogeneous ODE system… which method shall I use?

I am an undergrad statistics student and I am having troubles with non-homogeneous ODE systems. During my classes I went over just three methods for solving odes: Laplace transform, Fourier transform ...
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1answer
222 views

Solving a system of two second order ODEs using Runge-Kutta method (ode45) in MATLAB

I'm trying to solve the system of differential equations outlined in Von Karman's rotating disk flow. I got them into a system of ordinary differential equations: F(n), G(n), H(n) $$F'' = -G^2 + F^2 ...
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2answers
33 views

Find the first order system of linear equations

Regard the diff equation: $mϕ′′+aϕ′+(mg/L)ϕ=0$ $ϕ(0)=0.1$ $ϕ′(0)=0$ where $m=0.1,L=1,a=2,$ 1) Rewrite the second order diff equation as a system of first order linear equations. 2) What is the ...
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2answers
299 views

Undamped spring mass system

I have this study guide for an upcoming test for DE class I'm trying to figure out. A mass of 400 grams stretches a spring by 5 centimeters. (a) Find the spring constant k, the angular frequency ω, ...
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1answer
47 views

Can someone give me an example of how to work out an exact linear second order differential equation?

I have a theorem that states: If an equation $P(x)y''+Q(x)y'+R(x)y=0$ can be written in the form: $$[P(x)y']'+[f(x)y]'=0$$ then the equation is said to be exact. Now I need to expand and equate ...
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83 views

Solve for $y' + Py = ae^{bt}$

How do I solve $y' + Py = ae^{bt}$? My attempt: $y' + Py = ae^{bt}\Rightarrow Py - ae^{bt} + 1.\frac{\mathrm{d} y}{\mathrm{d} t}=0$, where $M(t,y)=Py - ae^{bt}$ and $N(t,y)=1$. $M_{y}=P$, and ...
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1answer
326 views

Solving a first order linear ODE and determining the behavior of its solutions

(a) Draw a direction field for the given differential equation. How do solutions appear to behave as $t → 0$? Does the behavior depend on the choice of the initial value $a$? Let $a_{0}$ be the value ...
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1answer
32 views

Division of differential equations

$$\frac{dx(t)}{dy(t)}=\frac{\alpha x(t) - \beta x(t) y(t)}{-\gamma y(t) + \delta x(t)y(t)}$$ How would one simplify this fraction? Maybe the chain rule could be of any use, but I don't see how.
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2answers
87 views

Second order ODE - why the extra X for the solution?

Assuming I have the following homogeneous ODE equation: $$a\cdot y'' + b\cdot y' + c \cdot y = 0$$ Why for $(b^2 - 4\cdot a\cdot c=0) \quad $,(meaning, when $m_1=m_2$) then the solution is: $$y = ...
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1answer
135 views

How many $f(x)$ are possible satisfying $f(x)=f'(x)$ and $f(0)=f(1)=0$.

Let $f:[0,1]\to\Bbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0,1)$ and $f(0)=f(1)=0$. Then the equation $f(x)=f'(x)$ admits how many solutions? The only solution ...
0
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1answer
114 views

Find the velocity of a flow

The question is: Find the velocity of the flow described by the velocity potential given in the polar coordinates $φ$$(r, θ)$ = $θ$, where $x = r cos θ$ and $y = r sin θ$, $r > 0, 0 ≤ θ < 2π$ ...