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L04 - Review Dierential equations Classication Complete...

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Review Differential equations, Classification, Com- plete Solution (Reduction to Integration) of FOLODE, Existence and Uniqueness Theorem for the general FOODE, Sepa- rable FOODE, Applications.
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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
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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 ψ ( x, y ) def = x 2 + xy 2 with 2 x + y 2 = ∂ψ ∂x and 2 xy = ∂ψ ∂y If x 7→ y ( x ) is a solution of ( * ) then ψ ( x, y ( x )) is constant. 2
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Indeed, ( x, y ( x ) ) dx = ∂ψ ( x, y ( x ) ) ∂x + ∂ψ ( x, y ( x ) ) ∂y dy ( x ) dx = (2 x + y 2 ( x )) + (2 xy ( x )) × dy dx = 0 , by ( * ) : ψ ( 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
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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 . Suppose there is a function ψ ( x, y ) such that M ( x, y ) = ∂ψ ∂x and N ( x, y ) = ∂ψ ∂y . ( E )
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