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Section 19.1 Ordinary differential equations

Suppose that \(y\) is an unknown function of variable \(t\text{,}\) with the property that \(y'(t) = 6t\) for all \(t\text{.}\) This qualifies as a differential equation, although a very simple one: since the right hand side is explicit, integration with respect to \(t\) tells us \(y(t) = 3t^2 + C\) where \(C\) is some undetermined constant. If we also know the value of \(y\) at some point (an initial condition), then \(C\) can be determined. So, indefinite integration (finding antiderivatives) is a special case of solving differential equations.

Suppose that \(y\) satisfies \(y'(t) = 6y(t)\) for all \(t\text{.}\) This is a more typical example of a differential equation. We cannot find \(y\) by integrating the right hand side because it is unknown. The solutions of this equation are \(y(t) = C e^{6t}\) where \(C\) is again an undetermined constant.

Usually the argument of unknown function is omitted, so the above equation would be written as \(y'=6y\text{.}\) Higher order derivatives may appear, for example \(y'' + 6y' - t^2 y = \cos t\text{.}\) Despite the appearance of terms like \(t^2\text{,}\) this differential equation is considered to be linear, because the unknown function and its derivative enter it linearly. The simplest equation of a nonlinear differential equation is \(y' = y^2\text{.}\)

A system of differential equations involves \(n\) equations with \(n\) unknown functions, for example

\begin{align*} y_1' \amp = 3y_1 + y_2 + t\\ y_2' \amp = t \sin(y_1)\cos(y_2) \end{align*}

This is usually considered as a single equation with an unknown vector-valued function \(\mathbf y\text{:}\)

\begin{equation*} \mathbf y' = \mathbf f(t, \mathbf y) \quad \text{where } \mathbf f(t, \mathbf y) = \begin{pmatrix} 3y_1 + y_2 + t \\ t \sin(y_1)\cos(y_2) \end{pmatrix} \end{equation*}

An equation (or system of equations) involving higher derivatives can be rewritten as a system that involves only the first derivatives. The trick is to consider \(y'\) as another unknown function, say \(z\text{,}\) and add the relation \(y' = z\) to the system. For example, \(y'' + 6y' - t^2 y = \cos t\) becomes the system

\begin{align*} y' \amp = z\\ z' \amp = - 6z + t^2 y + \cos t \end{align*}

Or in vector form, letting \(\mathbf y = \begin{pmatrix} y \\ z \end{pmatrix}\text{:}\)

\begin{equation} \mathbf y' = \begin{pmatrix} y_2 \\ -6 y_2 + t^2 y_1 + \cos t \end{pmatrix}\label{eq-ode-system-ex1}\tag{19.1.1} \end{equation}

We will not need theoretical methods of solving differential equations, but you should be able to rewrite an equation/system of higher order as a first-order system such as (19.1.1).

A terminological remark: all of the above are called ordinary differential equations (ODE) because there is only one independent variable, \(t\text{.}\) With more than one independent variable such equations involve partial derivatives and are called partial differential equations (PDE). We will not consider PDE in this course.