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Unformatted text preview: . 1 • 2) The function G of the hint satisﬁes the condition G ( x ,y ) = 0. Furthermore ∂G ∂y ( x ,y ) = 1 /g ( y ) 6 = 0 . Under these conditions the implicitfunction theorem guarantees that there exists a unique solution y = φ ( x ) of the equation G ( x,y ) = 0 reducing to y when x = x , in a suﬃciently small neighborhood of x . • 5) The righthand side is homogenous of degree zero and the solution, with y = xv , is given implicitly by Z v ds a + bs c + dss = ln x. • 10) Since by assumption ∂ ( pM ) ∂y = ∂ ( pN ) ∂x and ∂ ( qM ) ∂y = ∂ ( qM ) ∂y it follows immediately that the same relation holds with αp + βq in place of p or q . 2...
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 Spring '99
 Jordan
 Calculus, Derivative, Trigraph, Natural logarithm, arbitrary initial data

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