practice exam

# practice exam - MATH 216 Introduction to Differential...

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MATH 216 Introduction to Differential Equations Fall 2007 First Midterm Exam Name Section This examination booklet contains 8 problems on 9 pages. This is a closed book exam. No calculators are allowed. One 3 by 5 note card is allowed. Show all your work. No work, no points! Problem Possible score Your score 1 20 2 13 3 10 4 10 5 10 6 12 7 13 8 12 Total 100 1

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1. (10 Points) Solve the equation dy dx = y 1 + x 2 , y (1) = 1 Solution: dy y = dx 1+ x 2 2 pts 2 y = arctan x + c 3 pts 2 = π 4 + c 2 pts 2 y = arctan x + 2 - π 4 y = 1 / 2(arctan x + 2 - π 4 ) 2 3 pts 2
2. (10 Points) Consider x 2 dy dx + xy = 1 x , y (1) = 1 . (a) Determine the integrating factor ρ ( x ). ρ = x 2 pts (b) Rewrite the differential equation so that the left hand side is a perfect derivative. d ( x y ) dx = 1 x 2 2 pts (c) Solve for y ( x ). Solution: y = 1 x [ 1 x 2 + C ] y = - 1 /x 2 + C/x y = - 1 /x 2 + 2 /x 6 pts 3

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3. (15 Points) Consider the following initial value problem dx dt = 10 x - 5 x 2 x (0) = C (a) (4 pts) Find the equilibrium solutions and determine whether they are stable or not. The equilibrium solutions are x = 0 (unstable) and x = 2 (stable). 4 pts (b) (9 pts) Find the general and particular solution. Are there any singular solu- tions?
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