Lesson 12.1 - The Differential Operator

The differential operator $L$ is defined by:

$L [y] = \frac{d^{n} y}{d x^{n}} + \frac{p_{1} (x) d^{(n - 1)}}{d x^{(n - 1)}} + \dots + p_{(n - 1)} \frac{d y}{d x} + p_{n} y = (D^{n} + p_{1} (x) D^{n - 1} + \dots + p_{n - 1} D + p_{n}) y$

L := D^{n} y + p_{1} (x) D^{n - 1} + \dots + p_{n - 1} D y + p_{n} y

The equation can now be expressed as $L [y] = f (x)$

We can see that $L$ is a linear operator. That is, $L [c_{1} y_{1} + c_{2} y_{2} + \dots + c_{n} y_{n}] = L c_{1} y_{1} + L c_{2} y_{2} + \dots + L c_{n} y_{n}$

Consequently, if $y_{1}, y_{2}, \dots, y_{n}$ is a set of linearly independent solutions (fundamental set) to $L [y] = 0$ (the homogeneous equation) then $y = c_{1} y_{1} + c_{2} y_{2} + \dots + c_{n} y_{n}$ is also a solution.

Next Lesson: Lesson 12.2 - The Wronskian for nth-Order Differential Equations

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