131F06.pracprobs2.sols

131F06.pracprobs2.sols - Solutions for Practice Problems 2...

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Solutions for Practice Problems 2 for MTH 131: Fall 2006 1). Evaluate the following limits: (i) lim x 1 1 - x 2 1 - x (ii) lim x 0 x - sin( x ) x - tan( x ) Answers: (i) + (ii) - 1 / 2. 2). Solve the following initial value problem: dy dx = 2 x ( y + 1) , y (0) = 0 Answers: y = e x 2 - 1 3). Compute the indefinite integrals of these functions: (i) x - 5 / 2 (ii) x ( x - 1)(2 x +3) (iii) x x 2 + 2 x + 5 Answers: (i) - 2 / 3 x - 3 / 2 + C (ii) 1 / 5 ln( x - 1) + 3 / 10 ln(2 x + 3) + C 4). Solve the following initial value problem: dy dx = 2 xy 2 1 + x 2 , y ( x 0 ) = y 0 Find the interval of existence for the cases where y 0 > 0 and y 0 < 0. Answers: y = 1 y 0 - ln 1+ x 2 1+ x 2 0 - 1 . For y 0 > 0 interval of existence is x 2 < (1 + x 2 0 ) e 1 /y 0 - 1, inequality reversed for y 0 < 0. 5). By changing variables to y = u 1 - n , show that the ODE du dt = a ( t ) u + g ( t ) u n is transformed into a linear ODE for y . Answers: y = (1 - n ) a ( t ) y + (1 - n ) g ( t ).
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6). Here is a modified harvesting equation: N ( t ) = (1 - N ) N - h (1 + N 3 ) , N (0) = N 0 The harvesting rate depends on N , and increases as N increases ( h 0 is constant.) a) To find the equilibrium solutions, you must find the zeroes of the function f ( u ) = (1 - u ) u - h (1 + u 3 ). Rather than looking for exact values, argue as follows. Sketch the two graphs y = (1 - u ) u and y = h (1 + u 3 ) and look where they cross. Show that (i) when
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