DIFFERENTIAL EQUATIONS: THE BASICS
5 minute review. Remind students what a differential equation is, the difference between ordinary and partial, linear and non-linear, and what the order of a
differential equation is.
Class warm-up. Differentiate the following functions of x, and construct a linear
first-order ordinary differential equation containing a y-term whose solution is the
given function.
(a) y = x100 ,
(b) y = x ln x,
x
(c) y = ee ,
(d) y = x sin x cos x.
Problems. Choose from the below.
1. Differential equations for sin and cos.
(a) Show that, for constants a and b, the function y = a cos x + b sin x is a
solution to the differential equation
d2 y
+ y = 0.
dx2
(1)
(b) Find values for a and b in order to give a solution to Equation (1) satisdy
= 4 when x = 0.
fying y = 3 and dx
(c) For which a and b are those solutions to Equation (1) also solutions to
2
dy
the differential equation dx
+ y 2 = 1?
2. Powers.
(a) Avoiding backwards reasoning, show that, for constants a and n, the
function y = axn is a solution to the differential equation
2
d2 y
dy
dy
+ xy 2 = x
.
(2)
y
dx
dx
dx
(b) Hence find a solution to Equation (2) with y = 5 at x = 1 and y = 80 at
x = 2.
3. Exponentials.
(a) Find two values a1 and a2 of a such eax is a solution to the differential
equation
d2 y
dy
−7
+ 10y = 0.
(3)
dx2
dx
(b) For those values a1 and a2 , show that rea1 x + sea2 x is a solution to
Equation (3) for any constants r and s.
(c) Hence find a solution to Equation (3) satisfying y = 4 and
x = 0.
dy
dx
= 11 at
(d) Can you generalise this approach to solve any equation which is of the
d2 y
dy
form p dx
2 + q dx + ry = 0 (with p 6= 0)? Does it split into cases?
1
2
DIFFERENTIAL EQUATIONS: THE BASICS
dy
dy
y
For the warm-up, (a) is a solution to dx
= 100
x y, (b) to dx = x + 1, (c) to
dy
and (d) to dx
= y x1 + cot x − tan x . There are other options.
dy
dx
= yex
Selected answers and hints.
1. In part (c) you should find the condition that a2 + b2 = 1.
2. You should end up with the solution y = 5x4 .
3. You should get a1 and a2 equal to 2 and 5 (in some order), and then the
solution in part (c) is 3e2x + e5x . For part (d), we will cover the theory later
in the course.
For more details, start a thread on the discussion board.