Homework1Sol

# Homework1Sol - Homework 1 AMATH 383 Autumn 2009 Due Friday October 16 1(each 1 point For the following equations determine the order and if it is

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Homework 1 AMATH 383, Autumn 2009 Due: Friday, October 16 1. (each 1 point) For the following equations determine the order and if it is non-linear or linear: (a) y 0 - 5 y 2 + y = 0 , (b) y 00 y + y 0 + 2 y = 0 (c) 6 y - y 0 y 00 = 1 (d) y 0 ( x ) = (2 + x ) y ( x ) (e) d 4 x dt 4 + 2 t dx dt = exp( x ) (f) d dx ± y 00 ( x ) + xy ( x ) y 0 ( x ) = 1 Solution: (a) nonlinear, 1st order (b) nonlinear, 2nd order (c) linear, 2nd order (d) linear, 1st order(non-constant coeﬃcients) (e) non-linear, 4th order (f) non-linear, 3rd order 2. (4+4+7 points) Give the general solutions to the following equations: (a) y 00 = - 3 y 0 , (b) 6 y - y 0 y 00 = 1 (c) y 0 = 6 y - 4 y 2 - 2 Solution: (a) y ( x ) = C 1 + C 2 e - 3 x C 1 , C 2 = const (Ansatz e λx ) (b) y ( x ) = C 1 e - 3 x + C 2 e 2 x C 1 , C 2 = const (Ansatz e λx ) (c) separation of variables: Z d x = Z 1 6 y - 4 y 2 - 2 d y = Z ± 1 2 y - 1 - 1 / 2 y - 1 d y ln(2 y - 1) - 1 2 ln( y - 1) = x + C y ( x ) = ˜ Ce 2 x - 1 ˜ Ce 2 x - 2 3. (2+2 points) The following gives equivalent solutions to ODEs which written with diﬀerent integration constants. Give a relation between the constants. 1

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Homework, AMATH 383, Autumn 2009 (a) For y 0 = y (1 - y ) equivalent solutions are y ( x ) = C exp( x ) 1 + C exp( x ) = 1 1 + ˆ C exp( - x ) = exp( x + ˜ C ) exp( x + ˜ C ) - 1 (b) For y 00 + ω 2 y = 0 equivalent solutions are y ( x ) = C 1 exp( ω i x ) + C 2
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## This note was uploaded on 01/10/2010 for the course MATH 124 taught by Professor Walker during the Spring '08 term at University of Washington.

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Homework1Sol - Homework 1 AMATH 383 Autumn 2009 Due Friday October 16 1(each 1 point For the following equations determine the order and if it is

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