### Solve differential equation $-f'(x)= a_1 f(a_2 x+a_3)$ with $f(0)=1$.

How to solve the following differential equation\begin{align}-f'(x)= a_1 f(a_2 x+a_3),\end{align}where $f(0)=1$. I looked around I think this falls under the category of discrete delayed differential equations....Read more

### A fierce differential-delay equation: df/dx = f(f(x))

Consider the following set of equations:$$\begin{array}{l}y = f(x) \\\frac{dy}{dx} = f(y)\end{array}$$These can be written as finding some differentiable function $f(x)$ such that $$f^{\prime} = f(f(x))$$ For example, say $y(0) = 1$. Then $\left. \frac{dy}{dx} \right|_{x=0}$ is determined by the value of $y(1)$. The derivative at the $x=0$ had better be negative, otherwise by the time the function gets to 1, the value will be too great and will contradict the alleged value of hte derivative at $x=0$.Many years ago I tried various techniques to...Read more

### Characterisitic equation of delay differential equation

How do I derive the characteristic equation around a fixed point $x_0$, when the DDE is defined as $\tau\, dx(t)/dt=-x(t) + f(p-w\,x(t-d))$, where p,w,d $\in R$ are constants and $f:R\rightarrow R$ is a nonlinear function with $f(x)=1+tan(x)$ ?...Read more

### Euler Scheme of Delay Differential Equation

Given an ode $x' = f(t)$. Then a basic Euler discretization scheme yields $$x_{n+1} = x_n + h f(t_n).$$Now suppose you have a delay differential equation, say $x' = f(t-\tau)$, does it make sense to discretize as follows: $$x_{n+1} = x_n+ hf(t_n - \tau)?$$...Read more