# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### Estimation higher order

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