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

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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...
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