C251ex1(fa05)

# C251ex1(fa05) - NAME Math 251 ANSWER KEY to First Exam...

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Math 251 October 12, 2005 ANSWER KEY to First Exam NAME: Section #: There are 9 questions on this exam. Many of them have multiple parts. The point value of each question is indicated either at the beginning of each question or at the beginning of each part where there are multiple part Show all your work . Partial credit may be given. The use of calculators, books, or notes is not permitted on this exam. Please turn oﬀ your cell phone before starting this exam. Time limit 1 hour and 15 minutes. Question Score 1 24pt 2 10pt 3 10pt 4 8pt 5 12pt 6 10pt 7 10pt 8 8pt 9 10pt Total 102pt

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1. a. 2pt Consider the following diﬀerential equation y 0 = y + 2 t . Without solving it, determine the slope of the tangent line to the solution at the point (1 , 2). ANS. 2pt for plugging (1 , 2) into f ( t, y ) gives 2 + 2(1) = 4. Give 1pt for 1 + 2(2) = 5. b. 2pt Find the Wronskian W ( y 1 , y 2 ) of the functions y 1 = sin t and y 2 = cos t ANS. 2pt for sin 2 t + cos 2 t . 1pt for sin 2 t - cos 2 t . c. 2pt Suppose y 1 and y 2 are two solutions of the ODE y 00 + (sin t ) y 0 + y = 0. and suppose that their Wronskian by W ( y 1 , y 2 )( t ) is 2 at t = 0. Find W ( y 1 , y 2 )( t ) for any t . ANS. 2pt for (2 /e ) e cos t . 1pt for anything else having the form ? e cos t . For the initial value problems in parts d. through g. state whether or not one of our two existence and uniqueness theorems for ﬁrst order ODE’s guarantees a unique solution. If the answer is yes and the theorem provides an interval of existence, then state what the interval is without actually solving the equation . d. 2pt ( t + 2) y 0 + ( y - 1) 2 / 3 = 0 , y (3) = 0 ANS. 2pt unique solution guaranteed Remove 1pt if interval is given. e. 2pt ( t + 2) y 0 + ( y - 1) 2 / 3 = 0 , y (0) = 1 ANS. 2pt no guarantee applies f. 2pt ( t 2 + 2 t ) y 0 + y = 0 , y ( - 3) = 0 ANS. 2pt unique solution guaranteed to exist on ( -∞ , - 2) g. 2pt ( t 2 + 2 t ) y 0 + y = 0 , y (3) = 0 ANS. 2pt unique solution guaranteed to exist on (0 , )
In parts h. 2pt through j. assume that

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## This note was uploaded on 07/23/2008 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Penn State.

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C251ex1(fa05) - NAME Math 251 ANSWER KEY to First Exam...

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