204 
LINEAR EQUATIONS. 
Let y x be a particular value of y when X= 0, and assume 
y = Vi v < Substituting, we have 
the first line of which is by hypothesis 0. In the reduced 
equation let = u, then we have 
3 '*S +(2 ^ + ^‘ )m = X (19) ’ 
a linear equation of the first order for determining u. And 
this being found, we have 
v = judx +c. 
In the particular case in which X=0, we find from (19) 
(J e -f Xulx 
/ r~-/Xidx \ 
whence y = y x ^Cj 2 dx+C^j (20). 
3rdly. Linear equations are connected by remarkable analo 
gies with ordinary algebraic equations. 
This subject has been investigated chiefly by Libri and 
Liouville, who have shewn that most of the characteristic 
properties of algebraic equations have their analogies in linear 
differential equations. 
An algebraic equation can be deprived of its 2nd, 3rd ... r th 
term by the solution of an algebraic equation of the 1st, 
2nd, ... r — 1 th degree. A linear differential equation can be 
deprived of its 2nd, 3rd, ... r th term by the solution of another 
linear equation of the 1st, 2nd,... r — 1 th order. 
This may be proved by assuming y = vy v and properly de 
termining v so as to make in the resulting equation y x assume 
the required form.