SECOND YEAR CALCULUS - Introduction to First Order Equations (ODEs)

1 208MAE, Tutorial sheet 2



Tutorial sheet 2

Topics covered

 Introduction to ODEs
 First order ODEs
 Direct integration
 Separation of variables

Problems: basic level

2.01 Classify these ODEs (i.e. what order, linear/nonlinear, constant coefficients,
homogeneous/non-homogeneous?):

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑3 𝑦𝑦 𝑑𝑑𝑑𝑑 𝑑𝑑3 𝑦𝑦 𝜕𝜕2 𝑦𝑦 𝜕𝜕𝜕𝜕
(a) 𝑑𝑑𝑑𝑑
= 2𝑦𝑦 (b) 𝑑𝑑𝑑𝑑
− 3 𝑑𝑑𝑥𝑥 3 = 2𝑦𝑦 (c) 𝑑𝑑𝑑𝑑 − 3𝑦𝑦 𝑑𝑑𝑥𝑥 3 = 2𝑦𝑦 (d) 𝜕𝜕𝑥𝑥 2
− 3 𝜕𝜕𝜕𝜕 = 2𝑦𝑦

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑3 𝑦𝑦 𝑑𝑑2 𝑦𝑦
(e) 𝑑𝑑𝑑𝑑 − 2𝑦𝑦 = 3𝑥𝑥 2 (f) 𝑑𝑑𝑑𝑑 − 2𝑦𝑦 = 3𝑦𝑦 2 (g) 2 𝑑𝑑𝑑𝑑 − 0.5𝑥𝑥 𝑑𝑑𝑥𝑥 3 + 4 sin 𝑥𝑥 𝑑𝑑𝑥𝑥 2
= 2.5 𝑒𝑒 2−5𝑥𝑥

4
𝑑𝑑𝑑𝑑 𝑑𝑑3 𝑦𝑦 𝑑𝑑2 𝑦𝑦 𝑑𝑑𝑑𝑑 𝑑𝑑3 𝑦𝑦 𝑑𝑑2 𝑦𝑦
(h) 𝑑𝑑𝑑𝑑 − 0.5𝑦𝑦 𝑑𝑑𝑥𝑥 3 + 4 sin 𝑥𝑥 𝑑𝑑𝑥𝑥 2
= 2.5 𝑒𝑒 2−5𝑥𝑥 (i) 𝑑𝑑𝑑𝑑
− 0.5 �𝑑𝑑𝑥𝑥 3 � + sin 𝑥𝑥 𝑑𝑑𝑥𝑥 2
=0

𝑑𝑑𝑑𝑑 𝑑𝑑2 𝑦𝑦 𝑦𝑦
(j) y 2 − 3 y = sin x (k) 0.1 𝑑𝑑𝑑𝑑 − 0.5 𝑑𝑑𝑥𝑥 2 + 𝑦𝑦 cos 𝑥𝑥 2 = 𝑥𝑥

𝑑𝑑𝑑𝑑 𝑑𝑑2 𝑦𝑦 𝑥𝑥 𝑑𝑑𝑑𝑑 𝑑𝑑2 𝑦𝑦
(l) 0.1 𝑑𝑑𝑑𝑑 − 0.5 𝑑𝑑𝑥𝑥 2 + 𝑦𝑦 cos 𝑥𝑥 2 = 𝑦𝑦 (m) �𝑦𝑦 𝑑𝑑𝑑𝑑 − √𝑥𝑥 𝑑𝑑𝑥𝑥 2 + cos 5𝑥𝑥 = ln 𝑥𝑥

𝑑𝑑𝑑𝑑 𝑑𝑑2 𝑦𝑦 𝑑𝑑3 𝑦𝑦 𝑑𝑑𝑑𝑑 5𝜋𝜋 𝑑𝑑2 𝑦𝑦 𝑑𝑑3 𝑦𝑦 𝑦𝑦
(n) 𝑑𝑑𝑑𝑑 × 𝑑𝑑𝑥𝑥 2 − 𝑑𝑑𝑥𝑥 3 = 0 (o) 𝑑𝑑𝑑𝑑 + cos �12 𝑥𝑥� × 𝑑𝑑𝑥𝑥 2 − 𝑑𝑑𝑥𝑥 3 =
√𝑥𝑥+1

5𝜋𝜋 𝑑𝑑4 𝑦𝑦 𝑑𝑑𝑑𝑑 1 𝑑𝑑2 𝑦𝑦 𝑑𝑑3 𝑦𝑦 𝑑𝑑4 𝑦𝑦 𝑑𝑑𝑑𝑑 𝑑𝑑2 𝑦𝑦
(p) cos �12 � × 𝑑𝑑𝑥𝑥 4 + 0.1𝑥𝑥 3 − 3 𝑑𝑑𝑑𝑑 + 𝜋𝜋 × 𝑑𝑑𝑥𝑥 2 = 𝑑𝑑𝑥𝑥 3 (q) 𝑑𝑑𝑥𝑥 4 − 3 𝑑𝑑𝑑𝑑 + 𝑑𝑑𝑥𝑥 2 = cos(3 − 𝑦𝑦)

2.02 Solve ODEs using direct integration:
= (3 − t ) : find p in terms of t given the condition p = 3 when t = 2.
dy dp
(a) = 2 x (b)
2

dx dt
dy dy
(c) = 3 cos 2t (d) = 12e 2t if at t = 0 we know y = 0.
dt dt
2.03 Solve ODEs using separation of variables:
dy dy x dy dy
(a) = 2 y (b) =2 (c) = 2 xy (d) = 3x 2 e − y
dt dx y dx dx




208MAE (2020/2021)

, 2 208MAE, Tutorial sheet 2


2.04 Which of the following equations can be solved using direct integration and/or
separation of variables?
dy dy dy dy π
(a) = cos x sin y (b) sin y + x 2 = 0 (c) = x 2 + y (d) = 25 cos − x
dx dx dx dx 3
dy dy
(e) e x = y 3 (f) e x =1
dx dx



Problems: standard level


2.05 The streamlines of a fluid flow are given by
dy dy dy y + 1
(a) = e x (b) = C (c) = (x > -1, y > -1)
dx dx dx x + 1
dy y dy x
(d) = − (y > 0, x > 0) (e) =−
dx x dx y
Solve the differential equation and sketch the streamlines
2.06 Consider a tank full of water which is being drained out through an outlet. The height H
dH
(m) of water in the tank at time t (s) is given by = −0.0028 H . Given that when
dt
t = 0 s the height is 4 m find an expression for H in terms of t.
2.07 Solve these boundary value problems:
dy
(a) = x 8 y 12 if y (0) = 3
dx
dy
(b) e y = e 4 x −5 if y(0) = 5
dx
dy 3 x 2 + 4 x + 2
(c) = if y(0) = -1
dx 2( y − 1)
2.08 (from Math is Fun)
The more rabbits you have the more baby rabbits you will get. Then those rabbits grow
up and have babies too! The population will grow faster and faster, and the population
N at any time t is described by an ordinary differential equation:
dN
= rN
dt
where r is the growth rate of the population. This would result in exponential growth of
rabbit population.
A guy called Verhulst included k (the maximum population the food can support) to get:
dN  N
= rN 1 −  (the Verhulst equation)
dt  k 
where k is the maximum population the food can support.
If initially there were 5 rabbits, and k = 40, r = 2 - find and sketch the solution.

2.09 The bending moment of a beam of length l is given by
dM
= w( x + l )
dx
where w is the constant load. Find an expression for M in terms of x.

208MAE (2020/2021)