H2_130 - Math 415 Homework 1 2.2.27 ty(4 y y(0 = y0 3(a...

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Math 415 Homework 1 2.2.27 y 0 = ty (4 - y ) 3 , y (0) = y 0 (a) Determine how the behavior of the solution as t increases depends on the initial value y 0 . By looking at the direction field for this problem we see that if y 0 > 0 then y ( t ) 4 as t → ∞ , if y 0 < 0 y ( t ) → -∞ as t → ∞ , and if y 0 = 0 then y ( t ) = 0 for all t . 0 -7.5 -5 -2.5 0 2.5 5 7.5 1 -5 -2.5 2.5 5 (b) Suppose that y 0 = 0 . 5. Find the time T at which the solution first reaches the value 3 . 98. If we separate the equation we get Z dy y (4 - y ) = 1 3 Z t dt, which, using partial fractions, evaluates to - 1 4 ln | 4 - y | + 1 4 ln | y | = 2 3 t 2 + C. Or y 4 - y = Ke 8 3 t 2 . The initial condition tells us that K = 1 7 and solving for y gives us y ( t ) = 4 e 8 3 t 2 7 + e 8 3 t 2 . And a computer or calculator can tell us that this function first hits 3 . 98 when t 1 . 64764 = T 1
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2.4.17 y 0 = ty (3 - y ) 0 1 2 3 4 5 -5 -2.5 2.5 5 As t → ∞ , if y 0 < 0 y ( t ) → -∞ , if y 0 = 0 y ( t ) = 0, if y 0 > 0 y ( t ) 3. 2.3.20 A ball of mass 0 . 15 kg is thrown upward from a height of 30 m with a velocity of 20 m/sec. (a) To find the maximum height we must find a formula for the height, or at least the velocity. Neglecting
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