1
vote
0answers
20 views

Asymptotic Behaviour

We have the following nonlinear ODE: $f' = af-bg -(f+g)^k(f'(0) +g'(0)) + f'(0)$ $\big(G-T(x)\big) g' = -af+bg - g'(0)$ where $a,b,k,G$ are constants and $f'(0) , g'(0)$ are the initial conditions ...
2
votes
1answer
39 views
+50

Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
6
votes
1answer
115 views

How to solve the non-linear differential equation $y''=x-y^2$?

$y''(x)=x-y^2(x)$ I'm particularly interested in solutions when $x>0$. I've performed asymptotic analysis and reached the conclusion that solutions must behave as $\pm\sqrt{x}$ when $x\rightarrow ...
7
votes
1answer
193 views

First-term approximation for singular perturbation of ODE (with two turning points)

I'm reading "Introduction to Perturbation Methods" by Mark Holmes, and I came across an exercise that I don't know how to approach. As I decided to independently read this book, I have no ...
0
votes
0answers
22 views

Differential equation question to do with modelling gravity.

Hi I am given two models for the gravity of earth, the first is with $x_3$ normal to the earth and is $m\ddot{x}(t)=-mg(0,0,1)^T$ the other is with $x$ a distance vector from the centre of the ...
2
votes
1answer
38 views

Leading order approximation to differential equation

Find a leading order approximation to the solution of $\epsilon y'' + 2 y' + e^y = 0$, $y(0)=y(1)=0$ as $\epsilon \to 0$. I know there is a boundary layer near $x=0$ and not at $x=1$ so I can ...
1
vote
1answer
23 views

Estimation higher order

Consider non-dimensional differential equation for the height at the highest point is given by \begin{equation} h(\mu)= \frac{1}{\mu}- \frac{1}{\mu^2} \log_e(1+\mu) \end{equation} $0<\mu\ll 1.$ ...
2
votes
0answers
72 views

How to maximize speed of rest position approach of nonlinearly damped spring oscillator?

Inspired by comments to answer for this question: Suppose we have a system which is described by the equation $$\ddot x=-x+g(\dot x),$$ with initial conditions $x(0)=1$, $\dot x(0)=0$. If ...
3
votes
0answers
58 views

WKB and asymptotic behavior of second order differential equation

I want to study the large $x$ solution to a Riccati equation. After listening to the lectures on Mathematical Physics by Carl Bender, I have fallen in love with asymptotic analysis. But, by no means ...
1
vote
0answers
78 views

Method of dominant balance and perturbation theory

We know perturbation theory express the desired solution of differential equations in terms of a formal power series in some "small" perturbation parameters: $y=y_0+\epsilon ^1 y_1+\epsilon ^2 ...
4
votes
1answer
117 views

Asymptotic estimate of an oscillatory differential equation

Let $f\in C^1(\mathbb{R} ,\mathbb{R} )$ and satisfying the condition: $$ \forall x >0, \quad f(x)>0, \forall x<0 , \quad f(x)<0 $$ Let $(\alpha, \beta) \in \mathbb{R^2}$. ...
0
votes
0answers
42 views

Limits of generalized hypergeometric functions

For a (quite fiddly) asymptotic matching, I would like to be able to write the solution to \begin{equation} \frac{\mathrm{d}^5}{\mathrm{d}x^5}f(x) + \frac{10}{15^{1/2}} \frac{\mathrm{d}}{\mathrm{d}x} ...
1
vote
0answers
53 views

Boundary layer method

I am trying to solve the following differential equation using boundary layer method. $\psi ''(z) + \frac{1}{z} \psi'(z)(3 - \frac{4}{1+(\frac{z}{zc})^8})+ \frac{m^2}{1+(\frac{z}{zc})^8}\psi(z)=0$ ...
0
votes
0answers
30 views

Extending a power series?

I am studying a differential equation $$ y'(x)=g(x,y), $$ which has no analytic solution, however I have found that $y(x)$ is asymptotic to a series $$ f(x)=\sum_{k=0}^\infty a_kx^{-k} $$ as ...
1
vote
1answer
45 views

What exactly is a multi-scale operator and what does it do?

In a paper I'm reading at the moment we're concerned with a third order nonlinear ODE for which we know the solution near thr origin look something like an upside-down parabola crossing the y axis at ...
2
votes
1answer
107 views

Solving $f_n=\exp(f_{n-1})$ : Where is my mistake?

I was trying to solve the recurrence $f_n=\exp(f_{n-1})$. My logic was this : $f_n -f_{n-1}=\exp(f_{n-1})-f_{n-1}$. The associated differential equation would then be $\dfrac{dg}{dn}=e^g-g$. if ...
0
votes
1answer
29 views

Order of the solution to an IVP

Suppose you have the following IVP $$\dot{y}(t) = d_1 y^2 + d_2 \epsilon^{-1} y$$ with $y(0) = y_0$ and where $d_1$ and $d_2$ are two positive constants independent of $\epsilon$. What can you ...
6
votes
2answers
164 views

Estimating rate of blow up of an ODE

Suppose I have a differential equation $x'=f(x)$ and $f(x)>0$ grows super-linearly. I.e., $\lim_{|x| \rightarrow \infty} |f(x)|/|x| \rightarrow \infty$. Several related questions: (1) Can I ...
6
votes
0answers
177 views

Hints/Help studying an Abel Differential Equation

I want to know more than qualitative information about the Abel differential equation $\frac{dy}{dx}+y^3+x=0$. $\qquad ... \;(1)$ Since I donĀ“t know how to solve this and as far as could see, this ...
19
votes
2answers
996 views

please solve a 2013 th derivative question?

$ f(x) = 6x^7\sin^2(x^{1000}) e^{x^2} $ Find $ f^{(2013)}(0) $ A math forum friend suggest me to use big O symbol, however have no idea what that is, so how does that helping?
1
vote
1answer
150 views

Matched Asymptotic Expansion - Stretching Transofrmation

I'm having problems getting my head around a stretching transformation in the method of matched asymptotic expansions. I'm reading Introduction to Perturbation Methods (by Holmes) and he discusses the ...
2
votes
1answer
595 views

Method of dominant balance

Find the leading asymptotic behaviour as $x \rightarrow \infty$ of $$x^2y'' + (1 + 3x)y' + y = 0 $$ Can someone kindly explain me how to solve this problem? Im learning asymptotic analysis, and I ...
2
votes
0answers
177 views

Asymptotic Methods - Boundary Layer Problems

I am currently studying a course in Asymptotic and Perturbation Methods and we have recently started discussing "Boundary Layer problems". It is not clear to me, however, exactly what form "Boundary ...
2
votes
1answer
88 views

Differential equation leading behavior

Show that the solution of $x^{3}y''=y$ whose leading behavior as $x\rightarrow0$ is $e^{-2x^{-1/2}}$ is actually given by $x^{3/4}e^{-2x^{-1/2}}$. Do this by writing $y=e^{S(x)}$ and finding the ...
1
vote
2answers
175 views

Singular Perturbation Problem? Asymptotics?

I am in the midst of solving this equation $\epsilon \ddot{y}+\dot{y}+1-\frac{1}{(y+1)^{2}}=0$ with the boundary condition $y(0)=1$ and $\dot{y}(0)=-1$ and $\epsilon$ is small. To start off with, I ...
9
votes
1answer
1k views

How does a harmonic oscillator with nonlinear damping behave?

It is well known that for a harmonic oscillator with linear damping, $$\ddot x+c\dot x+x=0$$ with positive $c$, the amplitude of the oscillations decays exponentially when $c<2$. If it is higher ...
0
votes
1answer
80 views

WKB approximation question

I was reading some stuff on asymptotic analysis, but how do you get from the 1st line to the 2nd line? $y \sim \frac{1+x}{2\lambda}\exp\left(\frac{\lambda x}{1+x}\right) - ...
3
votes
1answer
392 views

Asymptotic behavior of a sequence given by a recurrence relation

Original problem is to determine asymptotic behavior of ${a_i}\left( t \right)$ as $t \to \infty $ given by recurrence relations ${a_1}\left( 0 \right) = 1$ ${a_1}\left( t \right) = \frac{{2t + ...
2
votes
1answer
286 views

Method of matched asymptotic expansions

Consider the equation $(x+1-\epsilon)\frac{dy}{dx}+(1-\frac{1}{4}\epsilon^2y)y=2(1-\epsilon x)$ with $y(1)=1$. I am interested in finding an asymptotic expansion for the inner solution so I put ...