DIFFERENTIAL EQUATIONS 
333 
EXERCISES 
1. A man swims across a river, always heading straight for the 
opposite hank. If the current is such that he is carried down stream 
with a velocity proportional to his distance from the nearer hank, 
find his path. Ans. A curve made up of two equal parabolic arcs. 
2. A circular turn-table rotates about its axis with uniform velocity. 
An ant steps on at the outer edge and crawls straight toward a light 
at the centre of the table. Find the path of the ant in space. 
Ans. r = a(l — cd). 
3. If the sun is setting in the west and the ant boards the turn 
table at its most easterly point and then always crawls straight 
toward the sun, show that the ant will describe an arc of a circle. 
4. If in Question 2 the light had been at a point fixed in space, on 
the circumference of the turn-table and diametrically opposite the 
point at which the ant steps on, obtain the differential equation of 
the path of the ant. 
II. Linear Equations of the Second Order, 
and Higher 
11. Elementary Theorems. The differential equation 
(1) 
dfy , p d n ~'y 
dx n dx 
^+- +P„-if x + P.y=li, 
in which the coefficients P u •••, P n , R, are given functions of x, 
which do not depend on y, is called a linear differential equation, be 
cause it is linear in y and its derivatives. If R = 0 : 
(2) 
*z. + P 1 *rh/+... 
dx n dx 11 " 1 
+ i\y = o, 
the equation is said to be homogeneous; otherwise, non-homogene ous. 
The homogeneous equations form far and away the more important 
class. 
Theorem I. If y l be a solution of the homogeneous linear dfferen 
tial equation (2), then cy x , where c is any constant, is also a solution. 
By hypothesis, y v satisfies equation (2), i.e. 
(3) ph+P l ffl + ... +P, yi = 0 
dx n 
dx n ~ l 
is a true equation. We wish to prove that cy l also satisfies equation 
(2), i.e. that