A person can consider a differential equation by determining an initial condition, as a differential equation is an equation that gives the derivative of an unknown function in terms of the function and the independent variable. A differential equation is considered differently depending on whether it is ordinary or partial and linear or non-linear.
For example, if y is the unknown function and t the independent variable, then an example of a differential equation would be dy/dt=f(y, t). Such a problem can be solved by finding the initial condition, such as y(0)=y0.
Often, differential equations enter applications as evolution equations. Thus, the independent variable is time, and the equation describes how some physical quantity evolves in time. If the right-hand side of the differential equation does not depend on y, then the differential equation can be solved using basic integration techniques. For example, if dy/dt=f(t), then y equals the integral of f(t), and the initial condition just specifies the constant of integration.
A delay differential equation uses a single variable for time. A stochastic differential equation is commonly used in physics and finance when the stochastic process is known. A differential algebraic equation is used to evaluate vectors. Different types of differential equations include linear equations, Bernoulli differential equations, second order differential equations and Laplace transforms.