336 
CALCULUS 
The coefficients, a ik and a k , are any continuous functions of x. The 
system is said to be homogeneous if a k = 0, k = 1, •••, n. 
Write out such a system (both non-homogeneous and homogeneous) 
for n — 2 and n — 3. 
By a solution of such a system is meant a set of n functions. 
Vk—fk{ x )^ k=l,”‘,n, 
which satisfy the given system. 
State and prove Theorems I and II for a homogeneous system, and 
the last Theorem of the text for a non-homogeneous system. 
11. Show that the linear differential equation 
x^ + Px^+Qy=B 
dx 2 (to J 
goes over by the substitution x = e l into the linear differential equa- 
tion 
d ll + (p _ l) Èf + Qy = R. 
dt 2 K ' dt 
Extend the theorem to linear differential equations of the w-th 
order. 
12. Constant Coefficients. We begin with the case of the homo 
geneous differential equation of the second order, 
(i) 
^ + 2 a— + fly = 0, 
dx 2 dx 
where a, /3 are given constants. It was early observed that the 
function e mx is a solution of this differential equation if m is a root 
of the quadratic equation 
(2) m 2 + 2 am + /3 = 0. 
For, compute the left-hand side of (1) when y = e mx . Here, 
^ = me mx , 
dx 
d 2 y __ 
dx 2 
m 2 e mx , 
and hence 
—^ + 2 « ^ + f3y = e mx (m 2 -f 2 am + /3). 
dx 2 dx 
Thus we see, for example, that the differential equation 
leads to the quadratic 
d Ll + 5 d y + 6y = 0 
dx 2 dx