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10Ma2aAnHw3

# 10Ma2aAnHw3 - (X3 Find all diï¬€erentiable functions f x...

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Ma 2a Fall 2010 (analytical track) PROBLEM SET 3 Due on Monday, October 18 I. Methods of integration of 1st order ODEs Problems 9.1(vi), 10.2, 10.4 (i) (X1) Solve the equation x 2 y 0 + 2 xy - y 3 = 0 . II. Applications of 1-st order ODEs Problem 9.3 (X2) Find orthogonal trajectories of the family of parabolas Π c = { x 2 = y + c } , c R . In other words, describe all smooth curves Γ such that if p Γ Π c for some c R , then the tangents of Γ and Π c at p are perpendicular.
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Unformatted text preview: (X3) Find all diï¬€erentiable functions f ( x ) satisfying the functional equation f ( x + y ) â‰¡ f ( x ) + f ( y ) 1-f ( x ) f ( y ) on some interval containing zero. III. Method of linearization (X4) Let Ï• Â± ( t ) denote the solution of the initial value problem Ë™ x = x + Â± ( t + x 2 ) , x (0) = cos Â±. Find approximately the value of Ï• Â± (1) for small Â± . 1...
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