Definitions

Ordinary Differential Equation

An ODE is an equation in the form $f(x,y,y^{\prime },y^{\prime \prime },...,y^n)=0$ where y is a function of x. Ordinary $\to$ $y$ is a function of $x$

Order

Order of an ODE is the order of it’s highest order derivative, so an order 3 has $y^{\prime \prime \prime }$

Normal Form

ODE is in normal form if the highest derivative is given as a function of the other derivatives and free variable. I.E. Is on it’s own side

$$y^{\prime \prime \prime }=a{y}^{\prime \prime }+f(t)$$

Initial Value Problem

A problem where you get an ODE packaged with initial values at a single point for all the variables except the highest derivative

Solution to IVP

A function $y=f(x)$ differentiable to the order of the ODE in the IVP defined in an open interval