Math 303 Review 2

# Math 303 Review 2 - Math 303 Review Sheet 2 Chapters 3-4 1...

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Math 303 Review Sheet 2 Chapters 3-4 1. A. Solve the following linear diﬀerential equations y 00 - y = 0 with y (0) = 3 ,y 0 (0) = 1. B. First solve the following linear diﬀerential equations 9 y 00 + 3 y 0 - 2 y = 0 ,with y (0) = 9 ,y 0 (0) = - 3, then describe the behavior the of solution. C. If y ( t ) is the solution of y 00 +3 y 0 +2 y = 0 with y (0) = 10 and y 0 (0) = - 20, then what is the limit of y ( t ) as t → ∞ ? 2.A. Determine the longest interval in which the given initial value problem is certain to have a unique twice dif- ferentiable solution. (a) ( x - 4) y 00 + 2 xy 0 + ( sinx ) y = 0 with y (0) = 1 and y 0 (0) = 2 (b) ( t 2 - 4) y 00 + ln ( t + 5) y 0 - ( t + 2) y = 0 with y (3) = 2 and y 0 (3) = - 4. B. Find two real solutions y 1 ( x ), y 2 ( x ) of equation ( y 0 ) 3 y 00 - y 3 = 0 and check that c 1 y 1 ( x )+ c 2 y 2 ( x ) is also a solution. 3. Determine whether a pair of functions are linearly dependent or linearly independent. A.
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