334 
CALCULUS 
(4) 
d n lcyA +Pl dn : 1 (ÆÙ + ... + p (cyf) = 0 
dx n dx n ^ v ' 
is a true equation. It is clear liow to draw the inference. 
Theorem II. If y x and y 2 be two solutions of the homogeneous 
linear differential equation (2), then their sum, y x + y2, is also a 
solution. 
The proof is similar to that of Theorem I, and is left to the 
student. 
Linearly Independent Functions. If n functions, f (x), • ••, /„(sc), 
are connected by an identical relation of the form : 
(5) cjx(x) + ••• + c n f n (x) = 0, 
where the c’s are constants not all 0, the functions are said to be 
linearly dependent. If no such relation between the functions exists, 
they are called linearly independent. 
Thus, for n — 3, the functions 
f (x) = sin x, /2 (x) = cos x, /3 (x) = sin (x -f a) 
are linearly dependent; for 
/i (x) cos a +/ 2 (x) sin a —/3 (x) = 0. 
For an arbitrary value of n the first n powers of x, namely, xf = 1, 
x l — x, x 2 , •••, as"" 1 , form a set of n linearly independent functions; 
for the function 
c 0 + c x x + c 2 as 2 + ••• +c n - x x n - 1 
vanishes identically when and only when each coefiicient is 0. 
Existence Theorem. It is shown in the theory of linear differen 
tial equations that, if the coefficients of the homogeneous linear 
differential equation (2) be continuous in an interval a ^ x ^ b, 
there exist n linearly independent solutions, y u •••, y n , each defined 
throughout the interval. 
From Theorems I and II it appears that the function 
(6) V = CiVv + c 2 y 2 + ••• -F c n y n 
is also a solution, where the c’s are any constants. 
And now it is shown, furthermore, in the theory of differential 
equations that, conversely, every solution of (2) in the above inter 
val can be written in the form (6). 
The Non-Homogeneous Equation (1). The solution of this equa 
tion can be referred to that of the corresponding homogeneous