Explanation Given information : The given system of equations is d x d t =- x + 3 y d y d t =- 3 x - y The given solution is Y 1 ( t ) = ( e - ( t − 1 ) sin 3 ( ( t − 1 ) ) , e - ( t − 1 ) cos ( 3 ( t − 1 ) ) ) Concept used : If the vector Y ( x ( t ) , y ( t ) ) is the solution of the given system of equations if the point ( x ( t ) , y ( t ) ) = ( f 1 ( t ) , f 2 ( t ) ) lies on the curve represented by the system of equations. That satisfies the equations. Calculations: Thegiven system of equations may be written as d x ( t ) d t = - x ( t ) + 3 y ( t ) .... (1) d y ( t ) d t =-3 x ( t ) - y ( t ) .... (2) Given solution is Y 1 ( t ) = ( e - ( t − 1 ) sin 3 ( ( t − 1 ) ) , e - ( t − 1 ) cos ( 3 ( t − 1 ) ) ) , i.e. x ( t ) = e - ( t − 1 ) sin 3 ( ( t − 1 ) ) and y ( t ) = e - ( t − 1 ) cos 3 ( ( t − 1 ) ) . We verify that x ( t ) = e - ( t − 1 ) sin 3 ( ( t − 1 ) ) and y ( t ) = e - ( t − 1 ) cos 3 ( ( t − 1 ) ) satisfy equation (1) and (2). Left-hand side of equation (1) : d x ( t ) d t = d ( e - ( t − 1 ) sin 3 ( ( t − 1 ) ) ) d t =3 e - ( t − 1 ) cos 3 ( ( t − 1 ) ) - e - ( t − 1 ) sin 3 ( ( t − 1 ) ) = e - ( t − 1 ) { 3 cos 3 ( ( t − 1 ) ) - sin 3 ( ( t − 1 ) ) } Right-hand side of equation (1): - x ( t ) + 3 y ( t ) = - { e - ( t − 1 ) sin 3 ( ( t − 1 ) ) } + { 3

4th Edition

ISBN: 9781133109044

Author: Blanchard, Devaney, and Hall

Publisher: ACME

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Chapter 2.6, Problem 4E

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Toverify: Y1(t)=(e- (t−1)sin3((t−1)),e- (t−1) cos(3(t−1))) is the solution of the given system of equations.

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Toverify : Y 1 ( t ) = ( e - ( t − 1 ) sin 3 ( ( t − 1 ) ) , e - ( t − 1 ) cos ( 3 ( t − 1 ) ) ) is the solution of the given system of equations.

Solution Summary: The author explains how the given system of equations is solved if the vector Y(x),y(t)) is the solution.

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Toverify: Y1(t)=(e- (t−1)sin3((t−1)),e- (t−1) cos(3(t−1))) is the solution of the given system of equations.

Chapter 2 Solutions