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Section 9.1 Definitions and Theorems

Subsection 9.1.1 Definitions

  1. A differential equation is an equation involving an unknown function and its derivatives.

  2. A differential equation is an ordinary differential equation if the unknown function depends on only one independent variable, otherwise it is a partial differential equation.

  3. The order of a differential equation is the order (number of derivatives taken) of the highest derivative appearing in the equation.

  4. The degree of a differential equation that can be written as a polynomial in the unknown function and its derivatives is the power to which the highest order derivative is raised.

  5. An \(n^{th}\) order differential equation is linear if it has the form

    \begin{equation} a_n(x) y^{(n)}(x) + a_{n-1}(x) y^{(n-1)}(x) + \dots + a_0(x) y(x) = b(x)\label{linearinhomo}\tag{9.1.1} \end{equation}

    A linear equation is homogeneous if \(b(x) = 0\) and inhomogeneous if \(b(x)\ne 0\text{.}\)

  6. A solution of a differential equation in the unknown function \(y\) and the independent variable \(x\) on the interval \(I\) is a function \(y(x)\) that satisfies the differential equation identically for all \(x\) in \(I\text{.}\)

  7. A particular solution is any one solution. The general solution is the set of all solutions.

  8. An initial value problem is a differential equation together with conditions (known as initial conditions) on the unknown function and its derivatives all at the same value of the independent variable.

  9. A boundary value problem is a differential equation together with conditions (known as boundary conditions) on the unknown function and its derivatives at more than one value of the independent variable.

Subsection 9.1.2 Notation

There are several different notations for derivatives in common use. You should be comfortable with all of them. Leibniz's notation for derivatives is:

\begin{equation} \frac{dy}{dx}, \qquad \frac{d^2 y}{dx^2}\qquad\frac{d^3 y}{dx^3}\qquad\dots\qquad\frac{d^n y}{dx^n}\tag{9.1.2} \end{equation}

Lagrange's notation for the same derivatives (wrt to the independent variable \(x\)) is:

\begin{equation} y^{\prime} \qquad y^{\prime\prime}\qquad y^{\prime\prime\prime} \qquad \dots \qquad y^{(n)}\tag{9.1.3} \end{equation}

Newton, who used time (\(t\)) as the independent variable, rather than \(x\text{,}\) used dots instead of primes:

\begin{equation} \dot{y} \qquad \ddot{y} \qquad \dots \qquad y^{(n)}\tag{9.1.4} \end{equation}

The equation for the general linear inhomogeneous differential equation, eqn.(9.1.1) above is long and messy looking. To simplify this equation, it is common to pull out all the derivative operators with their coefficients into a single differential operator acting on the unknown function \(y\) and denoting the big messy operator by the single caligraphic letter \(\LL\text{,}\) i.e.

\begin{equation} \LL\equiv a_n(x) \frac{d^{n}}{dx^{n}} + a_{n-1}(x) \frac{d^{n-1}}{dx^{n-1}} + \dots + a_0(x)\tag{9.1.5} \end{equation}

so that eqn.(9.1.1) becomes

\begin{equation} \LL y(x)=b(x)\text{.}\tag{9.1.6} \end{equation}