Unformatted text preview: f ( x,y ) = C . From ∂f ∂x = 2 x + y 3 , we see that f ( x,y ) = x 2 + xy 3 + φ ( y ). From ∂f ∂y = 3 xy 2 and f ( x,y ) = x 2 + xy 3 , we see that 3 xy 2 = 3 xy 2 + φ ( y ), so φ ( y ) = 0. We can choose φ ( y ) to be 0 as well. (In general, φ ( y ) could be some constant K , but this will turn out not to matter.) The solution to the diﬀerential equation is given by f ( x,y ) = C , or x 2 + xy 3 = C. (Note that replacing x 2 + xy 3 with x 2 + xy 3 + K on the lefthand side doesn’t change anything, as both constants can be moved to the righthand side.) The solution can further be rewritten as xy 3 = Cx 2 y 3 = C xx y = ( C xx ) 1 3 . 1...
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 Spring '07
 lee
 Derivative

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