L04 - Review Differential equations, Classification, Com-...

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Unformatted text preview: Review Differential equations, Classification, Com- plete Solution (Reduction to Integration) of FOLODE, Existence and Uniqueness Theorem for the general FOODE, Sepa- rable FOODE, Applications. Exact FOODE Example: Solve (2 x + y 2 ) + (2 xy ) dy dx = 0 , ( * ) rewritten abominably (2 x + y 2 ) dx + (2 xy ) dy = 0 . So what? 2 Exact FOODE Example: Solve (2 x + y 2 ) + (2 xy ) dy dx = 0 , ( * ) rewritten abominably (2 x + y 2 ) dx + (2 xy ) dy = 0 . So what? Observe that there is this function ( x,y ) def = x 2 + xy 2 with 2 x + y 2 = x and 2 xy = y If x 7 y ( x ) is a solution of ( * ) then ( x,y ( x )) is constant. 2 Indeed, d ( x,y ( x ) ) dx = ( x,y ( x ) ) x + ( x,y ( x ) ) y dy ( x ) dx = (2 x + y 2 ( x )) + (2 xy ( x )) dy dx = 0 , by ( * ) : ( x,y ( x ) ) = c ( I ) is an implicit solution of ( * ) . If we are lucky we can solve it for y ( x ) and get us an explicit solution. Let us try: x 2 + xy 2 = c = y 2 = ( c- x 2 ) /x = y ( x ) = q ( c- x 2 ) /x . 3 Scholium: Consider the FOODE M ( x,y ) + N ( x,y ) dy dx = 0 , ( * ) rewritten abominably as M ( x,y ) dx + N ( x,y ) dy = 0 ....
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This note was uploaded on 09/07/2010 for the course PHY 303L taught by Professor Turner during the Spring '08 term at University of Texas at Austin.

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L04 - Review Differential equations, Classification, Com-...

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