# Pts find the solution to initial value prob lem d 2 y

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Problem 8. 8. (5 pts) Suppose y ( t ) = c 1 e 2 t 1 1 + c 2 e t 0 1 , y ( 0 ) = 2 3 . (a) Find c 1 and c 2 . c 1 = c 2 = (b) Sketch the phase plane trajectory that satisfies the given initial condition. Which graph most closely re- sembles the graph you drew? ? (c) Is the solution curve headed toward or away from the origin as t increases?
d 2 y dt 2 16 dy dt + 64 y = 0 , y ( 0 ) = 1 , y ( 0 ) = 7 The solution is . C D
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(5 pts) Find the solution to initial value prob- lem A B
(incorrect) Correct Answers: 2 -5 A A Problem 9. 9. (5 pts) Consider the initial value problem y = 4 1 0 4 y , y ( 0 ) = 4 1 . (1) Find the eigenvalue λ , an eigenvector v 1 , and a gener- alized eigenvector v 2 for the coefficient matrix of this linear system. (2) Find the most general real-valued solution to the linear
system of differential equations. Use t as the indepen- dent variable in your answers.