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

learn more… | top users | synonyms (1)

2
votes
0answers
71 views

Differential Equation - $y'=5|y|^{4/5}, y(0)=0$

in the spirit of this question I ask about this one. $y'=5|y|^{4/5}, y(0)=0$ If $y> 0$ then $$y'=5|y|^{4/5}\iff y'=5^{-1}y^{4/5}\iff 5^{-1}y'y^{-4/5}=1\iff y^{1/5}=x+C\\ \iff ...
2
votes
1answer
67 views

Differential Equation - $y'=|y|+1, y(0)=0$

The equation is $y'=|y|+1, y(0)=0$. Suppose $y$ is a solution on an interval $I$. Let $x\in I$. If $y(x)\ge 0$ then $$y'(x)=|y(x)|+1\iff y'(x)=y(x)+1\iff \frac{y'(x)}{y(x)+1}=1\\ \iff \ln ...
2
votes
1answer
681 views

butcher tableau runge kutta methods

Hi I have had a go at this question- am i heading in the right direction? it would be much appreciated if someone could me Write the Butcher Tableau for the 1-stage $\theta$ method: $$U^n ...
2
votes
2answers
1k views

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. ...
2
votes
2answers
677 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. ...
1
vote
1answer
26 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 ...
1
vote
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 ...
1
vote
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} ...
1
vote
1answer
86 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 ...
1
vote
1answer
65 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 ...
1
vote
2answers
93 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 ...
1
vote
1answer
120 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 } ...
1
vote
2answers
89 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 ...
1
vote
2answers
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 ...
1
vote
1answer
300 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, ...
1
vote
2answers
120 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: ...
1
vote
1answer
72 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. ...
1
vote
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} ...
1
vote
3answers
349 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 ...
1
vote
2answers
391 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{ , } ...
1
vote
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 ...
1
vote
1answer
111 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!
1
vote
0answers
175 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 ...
1
vote
3answers
249 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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
2answers
11k 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 ...
1
vote
3answers
558 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 ...
1
vote
1answer
323 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 ...
1
vote
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 $s = -2-4i$ ...
1
vote
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 ...
1
vote
3answers
106 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.
1
vote
3answers
454 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 ...
1
vote
2answers
701 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: ...
1
vote
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? ...
1
vote
1answer
317 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 ...
0
votes
0answers
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 (*) ...
0
votes
0answers
82 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 ...
0
votes
1answer
42 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 ...
0
votes
3answers
42 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 $u=u(x)$, ...
0
votes
3answers
328 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 ...
0
votes
2answers
51 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 ...
0
votes
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 ...
0
votes
1answer
234 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 ...
0
votes
2answers
34 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 ...
0
votes
2answers
312 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, ...
0
votes
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 ...
0
votes
2answers
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 ...
0
votes
1answer
350 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 ...
0
votes
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.