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Unformatted text preview: Math 251 October 12, 2005 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 22pt 2 10pt 3 10pt 4 8pt 5 12pt 6 10pt 7 10pt 8 8pt 9 10pt Total 100pt 1. a. 2pt Consider the following differential equation y = y + 2 t . Without solving it, determine the slope of the tangent line to the solution at the point (1 , 2). b. 2pt Find the Wronskian W ( y 1 , y 2 ) of the functions y 1 = sin t and y 2 = cos t c. 2pt Suppose y 1 and y 2 are two solutions of the ODE y 00 + (sin t ) y + 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 . 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 ODEs guarantees a unique solution. If the answer is yes and the theorem provides an interval of existence, then state what the interval is...
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