Differential equations/Ordinary Differential Equations


 * See also 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. $$ \frac{d}{dx} f(x) $$. 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 $$ F = ma $$, and when combined with $$ \frac{d^2}{dt^2} x =\frac{d}{dt} v = a $$ led to the differential equation, $$ F = m \frac{d^2}{dt^2} x$$.

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).

An nth order differential equation is of the form $$ y^{(n)} = f(t,y,y',...,y^{(n-1)} $$). For example, when Newton's second law of motion, $$ F = ma $$, is applied to a moving object the resulting differential equation is $$ my''(t)= f(t,y(t),y'(t)) $$
 * 1) Law of Conservation of Mass: Rate of Mass In - Rate of Mass Out = Rate of Change of Mass content, $$ \Delta $$