FACULTY OF EDUCATION AND CURRICULUM STUDIES DEPARTMENT OF MATHEMATICAL SCIENCES Differential Equations MAT-302-2 1. A. Prove that the differential equation 𝑑𝑦 𝑎𝑡 + 𝑏𝑦 + 𝑚 = 𝑑𝑡 𝑐𝑡 + 𝑑𝑦 + 𝑛 Where a, b, c, d, m, and n are constants, can always be reduced to dy I dt = ( at + by) / ( ct + dy) if ad- bc ≠ 0. B. Solve the above equation in the special case that ad= bc. 2. Solve the following Des 𝑑𝑦 i. (1 + 𝑡 2 ) 𝑑𝑡 = 1 + 𝑦 2 ii. 𝑑𝑦 𝑑𝑡 iii. 𝑑𝑦 𝑑𝑡 = 𝑒 𝑡+𝑦+3 = 2𝑡 𝑦+𝑦𝑡 2 𝑦(2) = 3 The black rhinoceros, once the most numerous of all rhinoceros species, is now critically endangered. The black rhino, native to eastern and southern Africa, was estimated to have a population of about 100,000 around 1900. Because of hunting, habitat changes, competing species, and most of all illegal poaching, the number of black rhinos today is estimated to be below 3000. If the wild population becomes too low, the animals may not be able to find suitable mates and the black rhino will become extinct. There must be a minimum population for the species to continue. Suppose this minimum or threshold population for the black rhino is 1000 animals and that remaining habitant in Africa will support no more that 20,000 rhinos. How might we model the current population, P(t) of black rhinos? 3 a. For what values of P is the rhino population increasing? What can be said about the value of dP/dt for these values of P? b. For what values of PP is the rhino population decreasing? What can be said about the value of dP/dt for these values of P? c. Find a differential equation that models the population of rhinos at time t 4. Find the solution of the initial-value problem 𝑑𝑦 a. 3𝑡 2 𝑦 + 8𝑡𝑦 2 + (𝑡 3 + 8𝑡 2 𝑦 + 12𝑦 2 ) 𝑑𝑡 = 0 𝑦(2) = 1 𝑑𝑦 b. 4𝑡 3 𝑒 𝑡+𝑦 + 𝑡 4 𝑒 𝑡+𝑦 + 2𝑡 + (𝑡 4 𝑒 𝑡+𝑦 + 2𝑦) 𝑑𝑡 = 0 𝑦(0) = 1