Explanation Given information: Given that the function ( x 1 ( t ) , y 1 ( t ) ) = ( cos t , sin t ) is a solution of some first order autonomous system then the function ( x 2 ( t ) , y 2 ( t ) ) = ( − sin t , cos t ) is also a solution. Calculation: Let us consider d x d t = g ( x , y ) and d y d t = h ( x , y ) , g and h are independent of t since the system is autonomous. If x 1 ( t ) = x ( t − t 0 ) ; y 1 ( t ) = y ( t − t 0 ) ; Then d x 1 d t = d x d t ( t − t 0 ) = g ( x , y ) & d y 1 d t = d y d t ( t − t 0 ) = h ( x , y )

4th Edition

ISBN: 9780495561989

Author: Paul Blanchard, Robert L. Devaney, Glen R. Hall

Publisher: Cengage Learning

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Chapter 2, Problem 21RE

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Whether the function (x1(t),y1(t))=(cost,sint) is a solution of some first order autonomous system then the function (x2(t),y2(t))=(−sint,cost) is also a solution.

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Chapter 2, Problem 20RE

Chapter 2, Problem 22RE

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| Without evaluating the Legendre symbols, prove the following. (i) 1(173)+2(2|73)+3(3|73) +...+72(72|73) = 0. (Hint: As r runs through the numbers 1,2,. (ii) 1²(1|71)+2²(2|71) +3²(3|71) +...+70² (70|71) = 71{1(1|71) + 2(2|71) ++70(70|71)}. 72, so does 73 – r.)

Chapter 2 Solutions

Differential Equations

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Whether the function ( x 1 ( t ) , y 1 ( t ) ) = ( cos t , sin t ) is a solution of some first order autonomous system then the function ( x 2 ( t ) , y 2 ( t ) ) = ( − sin t , cos t ) is also a solution.

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Whether the function (x1(t),y1(t))=(cost,sint) is a solution of some first order autonomous system then the function (x2(t),y2(t))=(−sint,cost) is also a solution.

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Chapter 2 Solutions