LINEAR DIFFERENTIAL EQUATIONS. 
479 
Application of the Theorem (g.) to Integration. 
To apply the theorem (g-.), let us suppose that we have given the equation 
&c (23.) 
Let this by (g.) be transformed into 
We shall have 
= — 24^1+34^2— •••• 
02=^"— 34^1+ 
03 = 4 "®— .,..&c. 
The figure at the top of 4" denotes the place of the term in the given equation, that 
at the bottom marks the term in (g.) corresponding to it*. By going a little into the 
operations, the meaning of these symbols will be easily understood. 
If 0=0, or 4"=4"J — 4"2+ ... the equation thus transformed will be integrable once, 
and we shall have 
ST “ * X = O -j- ZtT O2M O3W + &C . 
If, moreover, d>i=0, it will be twice integrable. 
Let us take as a more particular example the equation 
4" + 'kWze + (4"! — 4 ^ 2 ) M = X, 
which is integrable because 0=0. Putting for 4"!, 4"2 their values, this becomes 
+ 4"^nTM + 9(4"^' — <p'4"®' — (p4"®") M=X (24.) 
The transformed equation will be 
or 
0TO2M-j-OiM=nT“*X ; 
which by substituting for w its value, is easily reduced to 
D«+ (|;+ (?4-.)->.-X. 
In order to abridge, make 
then the last becomes 
(D+^)m=(^4"®)-‘st-'X, 
which gives 
w=(D+^)“‘((p4"®)~'GT"'X, 
or 
* The figures at the top of 4" are not exponents of powers, but distinct functional marks. 
3 Q 2 
