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Final Exam - 2005F - SOLUTIONS

# Final Exam - 2005F - SOLUTIONS - Math 215/255 Final...

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Question 1: [12 marks] Solve each one of the following Frst-order initial value problems for a real-valued solution y ( t ) in explicit form. Also, determine the domain of deFnition for each solution. (a) y = y sin t + 2 te cos t , y (0) = 1. (b) 1 (1 t ) yy = 0, y (0) = 1. Solution: (a) Rewrite the eqn in the normal form y + ( sin t ) y = 2 te cos t . Thus the integrating factor is μ ( t ) = e R ( sin t ) dt = e cos t . Thus ( e cos t y ) = e cos t ( 2 te cos t ) = 2 t e cos t y = i (2 t ) dt = t 2 + C y ( t ) = e cos t ( t 2 + C ) . y (0) = 1 C = e y ( t ) = e cos t ( t 2 + e ) . It is deFne on ( −∞ , ). (b) This eqn is not exact but is separable. yy = 1 1 t 1 2 y 2 ( t ) = ln | t 1 | + C y 2 ( t ) = ln( t 1) 2 + C y ( t ) = ± r C ln( t 1) 2 . y (0) = 1 C = 1 y ( t ) = r 1 ln( t 1) 2 . It is deFne on the interval (1 e, 1). Page 2 of 14
Question 2: [12 marks] Answer “True” or “False” to the statements below. Put your answers in the boxes. (20 points) (a) Suppose the Wronskian of two functions f ( t ) and g ( t ) is W ( f, g )( t ) = t ( t 1) which is zero at t = 0 , 1. Then, f ( t ) and g ( t ) must be linearly dependent functions. False. (b) The Laplace transform of the initial value problem y ′′′ + y ′′ + y = 0 , y (0) = 1 , y (0) = 2 , y ′′ (0) = 3 yields Y ( s ) = ( s 2 + 3 s + 6) / ( s 3 + s 2 + s ).

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Final Exam - 2005F - SOLUTIONS - Math 215/255 Final...

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