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Unformatted text preview: 2 ( c an arbitrary constant). (b) Show that for any constants c 1 and c 2 x ( t ) = c 1 t 2 t c 2 t 2 t < is a solution of tx2 x = 0 on ( , ). (c) Why does this not contradict the existence and uniqueness theorem? (d) Show that any solution to tx2 x = 0 must satisfy x (0) = 0. Exam continues on other side 5. (10 points) Draw the phase portrait of dx dt = ( x 29)(1x ) 2 and label the equilibria as attractors, repellers or neither. 6. (14 points) The functions h 1 = t 2 ,h 2 = t1 and h 3 = 1 are solutions of (*) ( t 2 D 3 + 2 tD 22 D ) x = 0. YOU DO NOT HAVE TO VERIFY THIS. (a) Find the general solution of (*) and explain why this is the general solution. (b) Solve the nonhomogenous equation ( t 2 D 3 + 2 tD 22 D ) x = t 2 . 7. (10 points) Solve ( D1) 2 ( D + 3) x = 0 x (0) = 0 , x (0) = 1 , x 00 (0) = 0. 8. (20 points) Solve by any method (a) (9 D 21) x = t . (b) ( D 2 + 1) x = sin t . End of Exam...
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This note was uploaded on 10/29/2010 for the course MATH MATH38 taught by Professor Hasselblatt during the Fall '10 term at Tufts.
 Fall '10
 Hasselblatt
 Math, Differential Equations, Equations

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