Higher-Order DEs

Functional Notation

A functional is an operator or map whose domain and range are in the space of functions

H [f] = g if f and g are functions

A linear operator $L$ is a functional for which the following property holds:

$L [c_{1} f_{1} + c_{2} f_{2}] = c_{1} L [f_{1}] + c_{2} L [f_{2}]$ for constants $c$ and functions $f$

(like derivative)

A differential operator $D^{n} [y]$ is a linear operator that can be used to represent the $n^{t h}$ derivative of a dependent variable with respect to an independent variable

D y = D [y] = \frac{d y}{d x}

Linear operators can distribute just like variables

\begin{array}{r} D^{4} y - y = 1 \\ (D^{4} - 1) y = 1 \\ (D - 1) (D + 1) (D^{2} + 1) y = 1 \end{array}

Reduction of Order

Higher order DEs with constant coefficients

Same method from Second-Order Linear DEs

a_{n} y^{(n)} (x) + a_{n - 1} y^{(n - 1)} (x) + \dots + a_{1} y (x) + a_{0} y (x) = 0

Case: $r_{1}, r_{2}, \dots, r_{n}$ are real and distinct

General Solution is the linear combination of all of the corresponding exponential functions

y = c_{1} e^{r_{1} x} + c_{2} e^{r_{2} x} + \dots + c_{n} e^{r_{n} x}

Case: Some of $r_{1}, r_{2}, \dots, r_{n}$ are repeated

General Solution is the
Case: Some of $r_{1}, r_{2}, \dots r_{n}$ are complex

y = e^{α x} (c_{1} \cos (β x) + c_{2} \sin (β x))

Case: Repeated complex roots