Quiz 2

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Department
Mathematics & Engineering Courses
Course
MTHE 225
Professor
Gregory G Smith
Semester
Fall

Description
Quiz 2b Solutions ′′ ′ 1. Consider the equation y − 3y = 0. 3x (a) Show that y (x1 = 1 and y (x) 2 e are independent solutions. Evidently y is1a solution, as y = 0 and y = 0. For y , we have 2 = 3e ′ 3x 1 1 2 and y = 9e , sox 2 y − 3y = 9e 3x − 3 ▯ 3e3x = 0 2 2 These two solutions are linearly independent because one is not a scalar multiple of the other. More formally, we can compute the Wronskian ! 3x 1 e 3x W(y ,1 )(2) = det 3x = 3e ▯= 0 0 3e and so, since y an1 y are 2oth solutions on R, and the Wronskian never vanishes on R, these functions are linearly independent. ′ (b) Find a solution such that y(0) = 4 and y (0) = −2 We seek constants c ,c 1uc2 that 3x y = c 1 c e2 satisfies the given initial conditions. Now y = 3c e , so y (0) = −2 gives 2 c = −2/3, so y = c − 2e . Then y(0) = 4 gives 4 = c − , so c = 2 14. 2 1 3 1 3 1 3 Thus the particular solution we sought is 14 2 y = − e 3x
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