# hw2a - ..) 2. Solve 7.15 and 11.17 from the posted...

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HW2 – MAT 267 – Advanced Ordinary Differential Equations Due: October 18, 2010 1. An n -th order homogeneous linear differential equation is an ODE of the form y ( n ) + p 1 ( t ) y ( n - 1) + p 2 ( t ) y ( n - 2) + ··· + p n - 2 ( t ) y ±± + p n - 1 ( t ) y ± + p n ( t ) y = 0 , where y ( k ) denotes the k -th derivative of y . For simplicity we assume that all the functions p i ( t ) are continuous in all of R . (a) Recall that, in the case n = 2 , we deﬁned the Wronskian of two solutions y 1 and y 2 as W ( y 1 , y 2 )( t ) = y 1 ( t ) y ± 2 ( t ) - y ± 1 ( t ) y 2 ( t ) = det ± y 1 ( t ) y 2 ( t ) y ± 1 ( t ) y ± 2 ( t ) . Based on the second equality above, propose a generalization of the Wronskian to the general case W ( y 1 , y 2 , . . . , y n ) . (b) Using your deﬁnition of the Wronskian, prove the following theorem: Theorem. Let y 1 ( t ) , . . . , y n ( t ) be a collection of n solutions of the ODE, and assume that W ( y 1 , . . . , y n )( t 0 ) ± = 0 for some t 0 R . Then y 1 ( t ) , . . . , y n ( t ) form a fundamental set of solutions of the ODE. That is, any solution y ( t ) of the ODE can be written as a linear combination of the n solutions y 1 ( t ) , . . . , y n ( t ) . (Of course, you need to have the correct deﬁnition in (a) to prove the theorem.
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Unformatted text preview: ..) 2. Solve 7.15 and 11.17 from the posted exercises for HW 1. 3. Solve the following problems from the textbook: 2.3, 2.4, 2.12 (assuming λ ± = μ ), 3.7. 4. In class we saw the example x ± =-x y ± = y. The ﬁrst quadrant of the phase portrait (also in page 40 of the textbook) is reminiscent of the curves deﬁned by xy = c for c > . (a) Write down the solution of the system and use this to explain why the solution curves, with initial conditions in the ﬁrst quadrant, look like the curves mentioned above. (b) Now consider the system x ± =-ax y ± = by where a, b > , a ± = b . Write down the solution of this system and derive an equation satisﬁed by the solution curves. (c) Sketch the phase portrait of the system in (b). How does it differ from the one in (a)? Distinguish the cases a > b and a < b ....
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## This note was uploaded on 04/30/2011 for the course MECHANICAL MSC taught by Professor Dr.rahula during the Spring '10 term at University of Moratuwa.

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