131F06.pracprobs2.sols

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

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Unformatted text preview: 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 → 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 ) = y Find the interval of existence for the cases where y > 0 and y < 0. Answers: y = 1 y- ln 1+ x 2 1+ x 2- 1 . For y > 0 interval of existence is x 2 < (1 + x 2 ) e 1 /y- 1, inequality reversed for y < 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 ). 6). Here is a modified harvesting equation: N ( t ) = (1- N ) N- h (1 + N 3 ) , N (0) = N 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...
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This note was uploaded on 10/19/2011 for the course MATH 3354 taught by Professor Drager during the Fall '08 term at Texas Tech.

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

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