266StudyGuide - 1 If y(t = sin 2t is a solution of y 9y =...

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1. If y ( t ) = sin 2 t is a solution of y 00 + 9 y = f ( t ), then f ( t ) = A. sin 2 t B. A cos 3 t C. 0 D. 5 sin 2 t E. 13 sin 2 t 2. If y = y ( x ) is the solution to dy dx = 4 xy 2 + x 2 , y (0) = 4 , then y ( 2)= A. 4 B. 16 C. 1 D. 2 E. 2 2 3. The general solution to x 2 y 0 + 2 xy = e 5 x is 2
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4. The solution to the problem (2 xy + x 3 ) dx + ( x 2 + y 4 ) dy = 0 , y (0) = 1 is 5. The solution in implicit form of dy dx = x 2 + 3 y 2 2 xy is: 6. Which of the following best describes the stability of equilibrium solutions for the au- tonomous differential equation y 0 = y (4 - y 2 )? A. y = 0 unstable; y = 2 and y = - 2 both stable B. y = 0 unstable; y = 2 stable C. y = 0 and y = 2 both stable D. y = 0 stable; y = 2 unstable; y = - 2 stable E. y = 0 stable; y = - 2 and y = 2 both unstable 3
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7. Solve the initial value problem y 0 - y = e - t with y (0) = a . For what value(s) of a is the solution bounded (i.e., not tending to infinity as t + ) on the interval t > 0? 8. Initially a tank holds 50 gallons of pure water. A salt solution containing 1 3 lb of salt per gallon runs into the tank at the rate of 5 gallons per minute. The well mixed solution runs out of the tank at a rate of 2 gallons per minute. Let x ( t ) be the amount of salt in the tank at time t . Find a differential equation satisfied by x ( t ). (DO NOT SOLVE THE EQUATION) A.
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