fa11_mae260_hw6 - k have to be so that the mass does not go...

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MAE 260. Homework Assignment #6 Issued: Dec. 8, 2011 Due: Dec. 20, 2011 (in exam) 1. Suppose we have bacteria on a plate and suppose that we are slowly adding a toxic substance such that the rate of growth is slowing down. That is suppose that dP dt = ( 2 - 0 . 1 t ) P where P ( t ) denotes bacteria population at t . (a) Draw slope field and describe what could happen if the initial population P ( 0 ) = 1000. (b) Analytically find the population at t = 5. 2. A mass of 2 kilograms is on a spring with spring constant k newtons per meter with no damping. Suppose the system is at rest and at time t = 0 the mass is kicked and starts traveling at 2 meters per second. How large does
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Unformatted text preview: k have to be so that the mass does not go further than 3 meters from the rest position. 3. (a) Solve ˙ x = ± 1 ² 2 1-1 ² 2 ³ x where the initial condition is given as x ( ) = ± 1 2 ³ (b) Repeat part (a) with external forcing: ˙ x = ± 1 ² 2 1-1 ² 2 ³ x + ± 1 e-t ³ 4. Consider the linear system ˙ x = A x of two differential equations where A is a real coefficient matrix. What does the general solution of the system look like, if it is known that λ 1 = 1 + 2 i is an eigenvalue of A and v 1 = [ 1 i ] T is a corresponding eigenvector? 1...
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