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Unformatted text preview: Math 201 Assignment #1 Solutions Due: Sep 20 in tutorial Grade problems 3 and 4. 10 marks for each, total 20 marks for this assignment. 1. A cyclist moves along Ring Road, which is assumed to be a circle with radius r , at a constant angular speed > . Show that the x and y coordinates of the cyclists position P both satisfy the differential equation d 2 u dt 2 + 2 u = 0 . Suppose the initial phase is , the angle at time t is ( t ) = t + , so the x and y coordiates are x ( t ) = r cos( t + ) , y ( t ) = r sin( t + ) . We want to verify that both functions satisfy the equation. I will show for x ( t ) , y ( t ) is similar. Substitute x ( t ) into the equation, x =- r sin( t + ) , x 00 =- 2 r cos( t + ) =- 2 x x 00 + 2 x =- 2 x + 2 x = 0 that is, x ( t ) satisfy the equation, so it is a solution. Similarly, y ( t ) satisfy the equation, so y ( t ) is a solution. 2. Verify that the family of functions y = e- x 2 x e t 2 dt + c 1 e- x 2 , for value x > , is a solution of the differential equation y + 2 xy = 1 . We substitute y ( t ) into the equation and varify that it satisfy the equation, y = e- x 2 x e t 2 dt + e- x 2 x e t 2 dt !...
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This note was uploaded on 01/15/2012 for the course MATH 201 taught by Professor Steacy during the Winter '10 term at University of Victoria.
- Winter '10