Ordinary Differential Equation (ODE)

Partial Differential Equation (PDE)

Stochastic Differential Equation (SDE)

Integral Equation

ODEs involve only one independent variable, while PDEs involve two or more.

ODEs have constant coefficients, while PDEs have variable coefficients.

ODEs can be solved using analytical methods, while PDEs require numerical methods.

ODEs are used in statistics, while PDEs are used in physics.

$y' + p(x)y = q(x)$

$y'' + p(x)y' + q(x)y = 0$

$y''' + p(x)y'' + q(x)y' + r(x)y = 0$

$y^{(n)} + p_1(x)y^{(n-1)} + ... + p_n(x)y = 0$

Separation of Variables

Method of Undetermined Coefficients

Variation of Parameters

Laplace Transform

A sequence of random variables indexed by time or space.

A function that describes the probability distribution of a random variable.

A mathematical model for the evolution of a system over time.

A method for estimating the parameters of a statistical model.

To determine the rate of population growth.

To predict the carrying capacity of an environment.

To analyze the effects of environmental factors on population dynamics.

All of the above

A PDE used to model the evolution of probability distributions.

A method for solving SDEs.

A technique for parameter estimation in statistical models.

A type of ODE used in population genetics.

To determine the probability distribution of a stochastic process at a given time.

To calculate the transition probabilities between states in a Markov chain.

To estimate the parameters of a stochastic process.

To simulate the trajectories of a stochastic process.

It is a continuous-time stochastic process with independent increments.

It is used to model Brownian motion.

It is a key component in the theory of stochastic integrals.

All of the above

To estimate the survival function of a population.

To determine the hazard function.

To analyze the effects of covariates on survival.

All of the above

To model the arrival and departure processes in a queueing system.

To determine the waiting time distribution of customers.

To calculate the optimal number of servers in a queueing system.

All of the above

To model the dynamics of stock prices.

To price options and other financial derivatives.

To manage risk in financial portfolios.

All of the above

A PDE used to model the evolution of option prices.

A method for calculating the fair value of options.

A formula for determining the optimal exercise strategy for options.

All of the above

To model the spread of infectious diseases.

To estimate the basic reproduction number.

To design vaccination strategies.

All of the above

A compartmental model for infectious disease transmission.

A method for estimating the number of susceptible, infected, and recovered individuals in a population.

A technique for designing vaccination strategies.

All of the above