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de1solutions - Dierential Equations Test 1 Solutions Answer...

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Differential Equations Test 1 Solutions Answer FOUR questions. Show all working clearly. 1. Solve the initial value problem d y d x = y 2 + 2 xy 2 , y (0) = 1 . 2. Solve the initial value problem d y d x - y = 2 xe x , y (0) = 1 . 3. Find the value of A for which the differential equation (2 x + ye xy )d x + Axe xy d y = 0 is exact and solve it with this value. 4. Solve the differential equation d y d x + y x = y 2 x 2 . 5. A population N has intrinsic growth rate r > 0 and is harvested at constant rate h > 0 so that d N d t = rN - h. Determine N as a function of t and its initial value N 0 > 0. Show that there is a threshold value k such that if h < k then N grows without bound while if h > k then the population is driven to extinction. 1
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Solutions : 1. The given DE d y d x = (1 + 2 x ) y 2 is separable. Separate and integrate: - 1 y = d y y 2 = (1 + 2 x )d x = x + x 2 + k and impose the initial condition y (0) = 1 to find k = - 1. Conclusion: y = 1 1 - x - x 2 . 2. The given DE is linear (already normalized) with integrating factor exp( - 1d x ) = e - x so that d d x ( ye - x ) = 2 x and therefore ye - x = x 2 + k.
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