This preview shows pages 1–7. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Review Differential equations, Classification, Com plete Solution (Reduction to Integration) of FOLODE. Example: The FOLODE is y + 3 t y = t n . Review Differential equations, Classification, Com plete Solution (Reduction to Integration) of FOLODE. Example: The FOLODE is y + 3 t y = t n . We multiply by the integrating factor e integraltext 3 /t dt = e 3 ln t = t 3 and get t 3 ( y + 3 t y ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright = ( t 3 y ) = t n +3 , whence t 3 y ( t ) = t n +4 n + 4 + C and y ( t ) = t n +1 n + 4 + C t 3 . Well consider general FOODEs y ( t ) = f ( t, y ( t ) ) , y ( t ) = dy ( t ) dt , for a while. The function f of two vari ables t, y is called the coupling coefficient , vis. dy ( t ) = f ( t, y ( t ) ) dt 2 Direction Fields The slope is f ( t, y ) ( t, y ) t y A Direction Field 3 ( t, y ) t y The blue curves y = y ( t ) satisfiy y ( t ) = f ( t, y ( t )) The slope is f ( t, y ). at every point ( t, y ) of the t, y plane. A Direction Field plus two solutions 4 Existence and Uniqueness Theorem: Suppose the function f ( t, y ) and its partial derivatives...
View
Full
Document
 Fall '08
 Fonken
 Differential Equations, Equations

Click to edit the document details