# Search results

1. ### A Formula for this curve?

The action potential satisfy a 2nd order ODE according to the Hodgkin-Huxley model. See equation (30) of their seminal paper . This equation has no closed form solution, but can be solved numerically. . HODGKIN AL, HUXLEY AF. A quantitative description of membrane current and its...
2. ### I Good introductory book for chaos theory?

Nonlinear dynamics and chaos by Steven Strogatz is a classic and often the book used in introductory courses on the subject. It's exceptional well written and easily digestible. More advanced treatments of the subject depends on the direction you want to go in. Chaos is a big field with a lot of...
3. ### First order differential equation involving a square root

I would expect the equation to be $$\bigg(\frac{dR}{dt}\bigg)^2 = \frac{GM}{R}$$ if it where to describe the gravitational collapse of a non-relativistic star, as a fluid parcel located at distance ##R## is gravitational bound (the Viral theorem). But I might be wrong, I haven't really studied...
4. ### Solution of a parametric differential equation

No worries! It was me that read your original post in a hurry. Your differential equation in ##y^\prime## belong to a notorious difficult class of ODE's called Abel's nonlinear ODE's of the fist-kind. I haven't had the change nor time to study this class of ODE's, so I'm afraid that I can't...
5. ### Solution of a parametric differential equation

Your ODE is a second-order linear equation with constant coefficients. It is rather straightforward to solve, simply observe that you can write it in the following form $$\big(e^{kt}y^\prime\big)^\prime = - ae^{kt}.$$ Now you simply have to integrate twice.

18. ### I A simple equation with simple solution - how to solve it?

I agree. Therefore, let ##x = \sin\theta## and ##y = \sin\phi##, then the equation $$y-x=\sqrt{1-x^2} + \sqrt{1-y^2}$$ becomes \begin{align*} \sin\phi - \sin\theta &= \sqrt{1-\sin^2\theta} + \sqrt{1-\sin^2\phi} \\ &= \cos\theta + \cos\phi \end{align*} or (by the sum-to-product trig...
19. ### Showing that a function is surjective onto a set

I assume (since you are not precisely specifying it) that ##B(0,r)\subseteq\mathbb{R}^n## is the closed ball of radius ##r>0## centered at ##0##. You are given that ##0\in B(0,1)## is a fixed point of ##f##. This immediately tells us that there exist a point in ##x\in B(0,1)## for which ##f(x)...
20. ### I How can you prove the integral without knowing the derivative?

Here is an example of how you would go about computing a definite integral of ##f(x) = x^2## from "first-principle" without utilizing the fundamental theorem of calculus. I will only show the calculations for the right evaluated Riemann sum, as the calculations for the left evaluated Riemann sum...