# differential equation tutorial

```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
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
```