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# home4_09F - y °° − ± 3 x ² y ° ± 4 x 2 ² y = 0(a...

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Department of Mathematical Sciences University of Delaware Prof. T. Angell October 9, 2009 Mathematics 351 Exercise Sheet 4 Exercise 16: A particle moves in a straight line in such a way that its distance x from the origin at time t obeys the di ff erential equation ¨ x + ˙ x + x = 0. Assuming it starts from the origin with speed 30 ft./sec., what will be its distance from the origin, its speed, and its acceleration after π / 3 seconds? Exercise 17: Let ϕ o ( x ) = 1 , ϕ 1 ( x ) = x , ϕ 2 ( x ) = x 2 , 1 x 1. Show that this set of functions is linearly independent on the interval [ 1 , 1] by showing that the truth of the relation α o ϕ o + α 1 ϕ 1 + α 2 ϕ 2 = 0 implies that all the α ’s must be zero. Exercise 18: Consider the set { u 1 , u 2 } of functions where u 1 ( x ) = x 2 x 0 0 x > 0 u 2 ( x ) = 0 x < 0 x 2 x 0 (Sketch their graphs!) Show that this set of functions is linearly independent on R and that the Wronskian of these two function is identically zero everywhere. What can you conclude about the functions u 1 and u 2 ? Exercise 19: For x > 0 consider the di ff
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Unformatted text preview: y °° − ± 3 x ² y ° + ± 4 x 2 ² y = 0 . (a) Verify the y (1) ( x ) = x 2 is a solution. (b) Put y (2) ( x ) = u ( x ) x 2 and show that x 2 u °° + xu ° = 0. (c) Set v = u ° and show that xv =const. (d) Choose 1 as the constant and integrate to show that u ( x ) = ln ( x ) + d , d a constant. Verify that this indeed is a solution of the original diFerential equation. Exercise 20: This problem illustrates how small changes in the coeﬃcients of a diFerential equation may cause dramatic changes in the solution. (a) ±ind a general solution ϕ ( t ) of ¨ x − 2 a ˙ x + a 2 x = 0 for a a non-zero constant. (b) ±ind the general solution ϕ ° ( t ) of ¨ x − 2 a ˙ x + ( a 2 − ° 2 ) x = 0 in which ° is a positive constant. (c) Show that, as ° → 0, ϕ ° ( t ) does not in general approach the solution ϕ ( t ). DUE DATE: Wednesday, October 14 in class...
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