Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [delay-differential-equations]

Questions about delayed differential equations which are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.

Delay differential equations (DDEs) or, time-delay systems differ from ordinary differential equations in that the derivative at any time depends on the solution (and in the case of neutral equations on the derivative) at prior times.

The simplest constant delay equations have the form $$y'(t) = f(t, y(t), y(t-\tau_1), y(t-\tau_2),\ldots, y(t-\tau_k))$$ where the time delays (lags) $~\tau_j~$ are positive constants. More generally, state dependent delays may depend on the solution, that is $~\tau_i = \tau_i (t,y(t)) \ .~$

Systems of delay differential equations now occupy a place of central importance in all areas of science and particularly in the biological sciences (e.g., population dynamics and epidemiology). Interest in such systems often arises when traditional point wise modeling assumptions are replaced by more realistic distributed assumptions, for example, when the birth rate of predators is affected by prior levels of predators or prey rather than by only the current levels in a predator-prey model.

References:

https://en.wikipedia.org/wiki/Delay_differential_equation

http://www.scholarpedia.org/article/Delay-differential_equations

197 questions
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When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?

The question is: When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution? Edit I should add that $f:\mathbb R\to\mathbb R$ to the question.
user72273
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Exponential of the differential operator and DDE

Today, via Dan Piponi's answer on Quora, I learnt that $e^Df(x) = f(x+1)$ where $D$ is differential operator and $$e^D \triangleq \sum_{i=0}^{\infty} \frac{D^i}{i!}.$$ I was curious as to whether the differential equation $$\frac{df(t)}{dt} =…
Thiagarajan
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Find $f$ where $f'(x) = f(1+x)$

Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function such that $$f'(x) = f(1+x)$$ How can we find the general form of $f$? I thought of some differential equations, but not sure how to use them here. Thanks.
user 1591719
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What function satisfies $F'(x) = F(2x)$?

The exponential generating function counting the number of graphs on $n$ labeled vertices satisfies (and is defined by) the equations $$ F'(x) = F(2x) \; \; ; \; \; F(0) = 1 $$ Is there some closed form or other nice description of this…
Caleb Stanford
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How to solve differential equations of the form $f'(x) = f(x + a)$

What could one do to find analytic solutions for $f'(x) = f(x + a)$ for various values of $a$? I know that $c_1\sin(x + c_2)$ is solution when $a = \frac{1}{2}\pi$, and of course $c_1e^x$ when $a = 0$. For instance, is there a function satisfying…
jnm2
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3 answers

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three equilibrium points to equation (1) by…
Bernhard
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2 answers

Solving the differential equation $f'(x)=af(x+b)$

How does one find all the differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ which satisfy the equation $$ f'(x)=af(x+b),\quad \text{for}\quad a,b \in \mathbb{R}? $$ I see that functions of the form $\alpha e^{\beta x}$ and $\alpha…
Malper
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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…
Mark Fischler
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4 answers

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.
Lisa
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A functional-differential equation

Consider the functional differential equation $$4f\left(\frac x2-f(x)\right)+4~\epsilon~ f(x)f'\left(\frac x2-f(x)\right)=f(x)$$ for all $x\geq0$ together with the initial condition $f(0)=0$ and the additional constraint $f(x)\geq0$ for all $x$.…
Gerhard S.
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All the solutions of $f'(x)=f(x+\pi/2)$

Consider the following equation (with $f \in C^{\infty}(\mathbb{R})$): $$f'(x)=f(x+\pi/2)$$ This equation is satisfied by $f(x) = A\cos(x) +B\sin(x)$, for any $A,B \in \mathbb{R}$. Question: What are all the (other) solutions of this equation (if…
Sebastien Palcoux
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The simplest delay differential equation

I am trying to understand a bit about solutions of delay differential equations, so I tried analyzing one of the most simple ones: $$u'(t)=-\beta u(t-1), \text{and for } t\in [-1,0), u(t)=\phi(t), \text{for some known function $\phi$}.$$ One way to…
Cantor
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Connection between possibility of non-monotonic solutions to first-order delay differential equations and 1-d discrete dynamical systems?

Is there a connection between the possibility of non-monotonic solutions, including periodic or other oscillatory solutions, arising in first-order autonomous delay differential equations such as the delay logistic equation…
J W
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Solutions to $xf'(x) = f(x-1)$

Consider the differential equation: $$xf'(x) = f(x-1)$$ for $f : \mathbb{R} \to \mathbb{R}$. This has a solution given by $(1+x)$. However, this solution does not work for me, as I require a (non-zero) solution which satisfies $\lim_{x \to \infty}…
J. S.
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1 answer

Delay-differential equation

Consider the equation $$ f'(t)=\frac{f(t-b)}{t-b}$$ $f'(t)=\frac{df(t)}{dt}$ and $b$ is a constant. Does anyone know if this equation has a name, an analytic solution and how to find the solution? This is not a question about how to solve the…
bob
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