Advanced Engineering Mathematics, Lecture 2.2: Linear independence and the Wronskian

Advanced Engineering Mathematics, Lecture 2.2: Linear independence and the Wronskian.

Given n solutions of a linear homogeneous ODE, the Wronskian is the determinant of a matrix where the first row are the functions, the second row are their derivatives, and so on; the n'th row are their (n-1)'st derivatives. If the Wronskian is not the zero function, then the functions are linearly independent, and thus form a basis of the general solution. In this lecture, we mostly focus on 2nd order ODEs. Abel's theorem provides a formula for the Wronskian even if we can't solve the ODE. We use this formula to show how that can help us find a second solution when we only have one solution. This technique, called "reduction of order" entails turning our homogeneous 2nd order ODE into an inhomogeneous 1st order ODE. It works more generally to help find a final solution of an n'th order ODE when we only have n-1 solutions.

[TriThom]

the remark on the slide at around 12 minutes should say linearly dependent I believe.

[MarcelloZucchi]

At 5.35 you probably meant y''=4y