Ordinary Differential Equations
Ordinary differential equations (abbreviated ODEs) from this point forward should already be familiar to you from basic calculus. A large portion of this course is dedicated to simple rearrangements of equations into forms we would recall from our section on integration techniques in calculus 2.
An ordinary differential equation is a functional equation in terms of an independent variable and a dependent variable. The traditional notation implies that the independent variable is usually t for time, but it may also be concentration, mass, voltage, ect.
$$\frac{dy}{dt} = f(t, y)$$
ODEs can take many forms. We will adopt the process of writing the first derivative as $y^{\prime}$, the second derivative as $y^{\prime\prime}$, until it becomes onerous. If we were to consider the fifth derivative, instead of writing $ y^{\prime\prime\prime\prime\prime} $, we will write $y^{(5)}$ for the sake of clarity. We will also generally suppress the $t$ in $y(t)$ or $y^{\prime}(t)$ for the sake of writing a compact notation.
Some ODEs are very simple to solve and you will readily recognize their solution. Consider:
$$y’ = t \rightarrow y = \frac{t^2}{2} + C$$
Other ODEs may require you to learn wholly new techniques and, at first, certainly appear to have more opaque solutions:
$$y^{\prime\prime} + 4y^{\prime} + y = 0 \rightarrow y = C_1 e^{-2t} + C_2 te^{-2t} $$
Setting both constants equal to 1 for the sake of gaining an idea of what this solution might look like, we obtain
We also hopefully immediately realize by examining these two differential equations that the solutions are necessarily a family of of solutions and that additional conditions are needed to determine precise solutions.
I cannot stress the importance of ODEs sufficiently in terms of science and engineering. Any well-designed, quantitative experiment is expressible as an ODE. This is so important that I am going to say exactly the same thing again: Any well-designed, quantitative experiment is expressible as an ODE. Moreover, the shift to expressing experiments and theoretical frameworks in terms of differential equations has completely transformed how we do science; It has given us the modern world.
ODEs will also connect to a number of other areas beyond calculus. Exact equations and Laplace transforms are deeply connected to complex analysis. Series solutions connect us to number theory. Laplace transforms will presage Fourier Analysis. ODEs as a whole helped generate a modern field of study known as functional analysis. Perhaps the most obvious direction forward from ODEs is the study of partial differential equations (PDEs) which studies differential equations that are dependent on more than one variable.