Systems of Dierential Equations
1. Two species with populations x and y compete for their food supply. The equations describing the evolution of x and y are
x = ax by
y = cx + dy ,
where a, b, c, d are positive constants. Explain w
Models and Dierential Equations
1. Find the general solutions and sketch the solution curves of the dierential equations:
= ex ,
= sin x ,
= sinh x .
Answers. General solutions are:
(a) y = ex + C ,
Integration Techniques: I
1. Use substitutions to evaluate the following integrals
x1 log x dx
ex + 1
1 + x2
(a) Let u = log x. Then du = u (x) dx = x1 dx. Substituting,
x1 log x dx =
Integrals as Functions
1. When applying the Fundamental Theorem it is important to check that the conditions for the theorem are satised. In particular, discontinuities in the function
or its derivative can invalidate the formula. Co
Integration Techniques: 2
1. Find a reduction formula for the indenite integral
(1 + x2 )n
Hint: Take dv/dx = 1.
Solution. The case n = 1 is a standard integral: I1 =
= tan1 x + C.
(1 + x2 )
For any n integrate by parts
Linear Dierential Equations
1. Find the general solutions of
tx = t ,
+ 2y = ex ,
+ (1 2x)y = x2 .
2y = 3 ,
+ 2tx = 2t3 ,
Answers. General solutions are
= 0.10x + 1000,
(b) 23 years.
12. The spread of innovation (in agriculture and industry) has been successfully modelled by assuming that the rate of spread is proportional to both the number of
people already having adopted the new system and
Applications of Separable Equations
1. Fechners law states that
(An example of its application to describe the response of the eye to brightness
is given in Table 1.)
Answer. R = n ln S + C . The visual
(b) c(t) = c0 ekt .
10. In a dilute sugar solution the rate of decrease of concentration is proportional to
the concentration c. If c = 0.01 g/cm3 at time t = 0 and c = 0.005 g/cm3 at time
t = 4 hours, show that the concentration after 10 hours will 0.
Second-Order Dierential Equations
1. Find the general solution of
What is the particular solution satisfying y(0) = y (0) = y (0) = y (0) = 1?
Answer.General solution: y = Ax3 + Bx2 + Cx + D.
1. A mothball initially has radius 0.5 cm and slowly evaporates.
(a) If V denotes the volume of the mothball and r the radius, use the chain rule
for dierentiation to show that
= 4r2 .
(b) Suppose that the ra