Differential equations/Ordinary Differential Equations

Work in progress

For engineers and scientists, your introduction to a differential equation probably occurred in your Calculus I class, where you were introduced to the derivative of a function (i.e. ${\displaystyle {\frac {d}{dx}}f(x)}$. At the same time you were taking introductory physics where concepts such as Newton's second law of motion (for linear motion) was presented as ${\displaystyle F=ma}$, and when combined with ${\displaystyle {\frac {d^{2}}{dt^{2}}}x={\frac {d}{dt}}v=a}$ led to the differential equation , ${\displaystyle F=m{\frac {d^{2}}{dt^{2}}}x}$ .

Similarly many fundamental laws of science are expressed as differential equations:

1. Law of Conservation of Mass: Rate of Mass In - Rate of Mass Out = Rate of Change of Mass content
2. Law of Conservation of Energy: Rate of Energy In - Rate of Energy Out = Rate of Change of Energy content

Each of these represents the change in a quantity (dependent variable) with respect to an independent variable (such as time).

1. Law of Conservation of Mass: Rate of Mass In - Rate of Mass Out = Rate of Change of Mass content, ${\displaystyle \Delta }$

An nth order differential equation is of the form ${\displaystyle y^{(n)}=f(t,y,y',...,y^{(n-1)}}$). For example, when Newton's second law of motion, ${\displaystyle F=ma}$, is applied to a moving object the resulting differential equation is ${\displaystyle my''(t)=f(t,y(t),y'(t))}$