Lesson 12.2 - The Wronskian for nth-Order Differential Equations

The Wronskian of functions $f_{1}, f_{2}, f_{3}, \dots, f_{n}$ is

W (f_{1}, f_{2}, \dots, f_{n}) = | \begin{matrix} f_{1} & f_{2} & \dots & f_{n} \\ f_{1}^{'} & f_{2}^{'} & \dots & f_{n}^{'} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ f_{1}^{n - 1} & f_{2}^{n - 2} & \dots & f_{n}^{n - 1} \end{matrix} |

If $y_{1}, y_{2}, \dots, y_{n}$ are $n$ solutions to the homogeneous equation $L [y] = 0$ , where $p_{1}, p_{2}, \dots, p_{n}$ are continuous functions of $x$ and $W (f_{1}, f_{2}, \dots, f_{n}) \neq 0$ , then every solution can be expressed in the form $y = c_{1} y_{2} + c_{2} y_{2} + \dots + c_{n} y_{n}$ , where $c_{1}, c_{2}, \dots, c_{n}$ are constants.

Next Lesson: Lesson 12.3 - Another Theory of Linear Differential Equations (Theorem 2)

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