rev3 - Sample Problems for Exam Three 1. Solve the system x...

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Sample Problems for Exam Three 1. Solve the system x 0 = x - 4 y ; y 0 = x + y 2. Factor the di±erential equation y 00 - 3 y 0 +2 y = x into two first order equations and solve. 3. Find the general solution of y 000 - 3 y 0 + 2 y = 0. 4. What does it mean to say that y 1 , y 2 and y 3 are independent? Show that x , x 2 - 1 and x 2 - 4 are independent using the definition. 5. Find the general solution of y ( viii ) - y ( vii ) - y ( iv ) + y ( iii ) = 0, given that r 8 - r 7 - r 4 + r 3 = r 3 ( r - 1) 2 ( r + 1)( r 2 + 1). 6. Use Undetermined Coe²cients to solve y 000 - y 00 + 4 y 0 - 4 y = cosx . 7. Find a fifth order linear di±erential equation with general solution y = c 1 e 2 t + c 2 e - 2 t + c 3 te - 2 t + c 4 cos 2 t + c 5 sin 2 t . 8. (a) Find the general solution to the Cauchy-Euler equation L [ y ] = x 3 y 000 - x 2 y 00 + 2 xy 0 + - 2 y = 0. (b) Find a particular solution of L [ y ] = x 3 by Undetermined Coe²cients. (c) Find the solution of
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