sol4 - MATHEMATICS 3161 Fall 2009(2009.9 2009.12 Assignment...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATHEMATICS 3161: Fall 2009 (2009.9 - 2009.12) Assignment # 4 (Due Date: Nov. 4) 1. The functions y 1 ( t ) = parenleftBigg 5 1 parenrightBigg , y 2 ( t ) = parenleftBigg 2 e 3 t e 3 t parenrightBigg are known to be solutions of the homogeneous linear system y ′ = Ay , where A is a real 2 × 2 constant matrix, a) Verify that the two solutions form a fundamental solution set; b) what is trA ? (Hint: Use Abel’s theorem), c) show that Ψ( t ) satisfies Ψ ′ = A Ψ, where Ψ = [ y 1 , y 2 ] = parenleftBigg 5 2 e 3 t 1 e 3 t parenrightBigg , d) use the observation of part c) to determine the matrix A (Hint: A = Ψ ′ Ψ − 1 ). Are the results of part (b) and (d) consistent? Solution : a) Since W [ y 1 , y 2 ] = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 5 2 e 3 t 1 e 3 t vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = 3 e 3 t negationslash = 0 for any t , so { y 1 , y 2 } is a fundamental solution set; b) From W ′ = tr ( A ) W , we have 9 e 3 t = tr ( A ) W , then tr ( A ) = 3; c) Since y 1 , y 2 are solutions of y ′ = Ay , so y ′ 1 = Ay 1 , y ′ 2 = Ay 2 . Therefore Ψ ′ = [ y ′ 1 , y ′ 2 ] = [ Ay 1 , Ay 2 ] = A [ y 1 , y 2 ] = A Ψ; d) From Ψ ′ = A Ψ, we can obtain A = Ψ ′ Ψ − 1 . Since Ψ = parenleftBigg 5 2 e 3 t 1 e 3 t parenrightBigg , Ψ − 1 = 1 3 e 3 t parenleftBigg e 3 t − 2 e 3 t − 1 5 parenrightBigg = parenleftBigg 1 3 − 2 3 − e- 3 t 3 5 e- 3 t 3 parenrightBigg then A = Ψ ′ Ψ − 1 = parenleftBigg 0 6 e 3 t 0 3 e 3 t parenrightBiggparenleftBigg 1 3 − 2 3 − e- 3 t 3 5 e- 3 t 3 parenrightBigg = parenleftBigg − 2 10 − 1 5 parenrightBigg . It’s easy to see that tr ( A ) = − 2 + 5 = 3 which is consistent with the result in b). 2. Consider the functions y 1 ( t ) = parenleftBigg t 2 2 t parenrightBigg , y 2 ( t ) = parenleftBigg e t e t parenrightBigg . a) Compute the Wronskian of y 1 ( t ) and y 2 ( t ); b) In what intervals are y 1 ( t ) and y 2 ( t ) linearly independent? c) What conclusion can be drawn about the coefficients in the system of homogeneous differ- ential equations satisfied by y 1 ( t ) and y 2 ( t )? d) Find this system of equations and verify the conclusions of part c). Solution : a) W [ y 1 , y 2 ] = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle t 2 e t 2 t e t vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = ( t 2 − 2 t ) e t b) At t = 0 and t = 2, W [ y 1 , y 2 ] = 0, so the vectors y 1 and y 2 are linearly independent on D = ( −∞ , 0) ∪ (0 , 2) ∪ (2 , ∞ ). c) Since y 1 and y 2 are linearly dependent at t = 0 and t = 2, so at least one of the coefficients in the system of homogeneous differential equations must be discontinuous at...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern