Introduces the Riccati Equation, and explains the various solution methods including: 1) Solution through transformation into a second order linear equation, 2) Solutions through known particular solutions, ranging from one known particular solution to four known particular solutions, and 3) Solution though separable form including Rao and Ukidave's, and Allen and Stein's transformations. Also shows how to transform the Riccati equation into reduced form, and canonical form. Contents timeline below for easy navigation:
00:00 - Introduction and historical context
06:33 - Contrasting Riccati equation against other simple ODEs such as Bernoulli
07:26 - Outline of the Solution methods
11:30 - 1) Solution via Transformation to linear equation
18:39 - 1-a) Alternative transformation methods
21:19 - 1-b) Example method for solving the transformed linear equation
26:11 - 2) Solution of Riccati when a particular solution is known
31:35 - 3) Solution when 2 particular solutions are known
36:35 - 4) Solution when 3 particular solutions are known
38:20 - 5) Solution when 4 particular solutions are known
40:04 - 6) Special form of Riccati Equation with easier solution method
40:22 - 6-a) Transformation to reduced form
42:48 - 6-b) Separable form
47:39 - Summary- solution recipe! Including Polynomial coefficients
8 июл 2024
