Diffeq Final Equations to Know

Differential Equation

A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives.

Order

The order of a differential equation is the largest derivative present in the differential equation.

ODE

A differential equation is called an ordinary differential equation, abbreviated by ode, if it has ordinary derivatives in it.

PDE

A differential equation is called a partial differential equation, abbreviated by pde, if it has partial derivatives in it.

A linear differential equation

Any eqn that can be written in this form

Solution to differential equation

A solution to a differential equation on an interval α<t<β is any function y(t) which satisfies the differential equation in question on the interval α<t<β.

Initial Equations

Initial Condition(s) are a condition, or set of conditions, on the solution that will allow us to determine which solution that we are after.

IVP

An Initial Value Problem (or IVP) is a differential equation along with an appropriate number of initial conditions.

Interval of validity

The interval of validity for an IVP with initial condition(s)

y(t0)=y0and/or y(k)(t0)=yk

is the largest possible interval on which the solution is valid and contains t0.

General Solution

The general solution to a differential equation is the most general form that the solution can take and doesn’t take any initial conditions into account.

Actual Solution

The actual solution to a differential equation is the specific solution that not only satisfies the differential equation, but also satisfies the given initial condition(s).

Explicit Solution

An explicit solution is any solution that is given in the form y=y(t). In other words, the only place that y actually shows up is once on the left side and only raised to the first power.

What is the format for a linear equation

dy/dt+p(t)y=g(t)

Integrating Factor

μ(t)=e^∫p(t)dt+k

Solution to a linear ODE

Seperable Equation

Exact Differential Equation