0
votes
1answer
44 views

Quadratic differential equation - closed form solution?

Can a solution for x(t) be found from the following real valued differential equation $$a\frac{d}{dt}\!(x(t))^2 + x(t) +b\frac{d}{dt}\!(y(t)) = 0$$ in terms of only y(t), it's integrals or ...
6
votes
2answers
254 views

Transition time in a Lotka-Volterra system

I am working with a set of real-valued ordinary differential equations based on the Lotka-Volterra competition equations: $$\begin{align} \dot{a_1} & = a_1 \left( 1 - a_1 - 2 a_2 \right) \\ ...
0
votes
1answer
21 views

Is there a closed form solution for the motion of a particle with friction?

I am trying to find a solution to Newton's equation of motion $ \boldsymbol{F} = m \boldsymbol{\ddot{r}} $ assuming a constant force $ \boldsymbol{F} $ but accounting for kinetic friction which is a ...
0
votes
1answer
57 views

Can every recurrence relation be solved?

Motivation A possible way to solve an ODE is to express the solution as: $y= \sum_{n=0}^\infty a_nx^n$. We substitute in the ODE and then calculate the coefficients $a_n$. For example, $y''+y=0$ ...
2
votes
2answers
95 views

How to solve $a \frac{d^2 y}{d x}+b \frac{d y}{d x} = f(y)$?

Let $a,b$ be real numbers and $y$ is a function of $x$. $f$ is a given function. How to solve the ODE : $a \dfrac{d^2 y}{d x}+b \dfrac{d y}{d x} = f(y)$ ? Can it be done in closed form ?
3
votes
0answers
108 views

A photon in expanding Universe (a snail on a tree)

I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
12
votes
1answer
403 views

Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
17
votes
1answer
348 views

Addition formula for $f_n(x+y)$ in closed form.

$n$ is a positive integer. $$f_n(x)^n+\left(\frac{df_n(x)}{dx}\right)^n=1$$ $f_n(0)=0$, $f_n'(0)=1$ then I am looking for the addition formula for $f_n(x+y)$ in closed form. if $n=1$ then ...
1
vote
1answer
163 views

Closed-form solution for this system of ODEs

I am trying to solve the following system (derived from a Michaelis-Menten kinetics model for an enzymatic chemical reaction): $$\dot{y}_a = r_p x_a - \lambda_p y_a$$ $$\dot{x}_b = \frac{\alpha_0 + ...