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 $\frac{d^2}{d{t}^2}x=\frac{d}{dt}v=a$ led to the differential equation , $F=m\frac{d^2}{d{t}^2}x$ .

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, $\mathrm{\Delta }$

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