LINEAR DIFFERENTIAL EQUATIONS. 
473 
After i transformations, we have 
Pi§^l ^m+i— 1 '^m 
If pi=0, ov p=ia^n—m—^-^^, the equation is solved, and we have 
Ui = T-W-li T-'X. 
From this we easily find u. The success of the method, it will be seen, depends 
upon the value of p. 
Other cases and solutions may be seen in the Philosophical Magazine in the article 
before mentioned, but they cannot be given here. 
Reduced to the ordinary form, (18.) becomes 
If we were to substitute for (p and § their values, this would become very compli- 
cated. We see that it differs considerably in form from the preceding examples 
which have been thus reduced. 
There is one equation bearing some analogy to that which has just been solved 
which deserves to be noticed here, although its solution must be effected by a very 
different process. It is 
VnU+pf^^-^^u=X (19.) 
By (c.) this may be put under the form 
Make and the last, by substituting this value, will become 
-\-pV = X. 
Now let T„g’’"“”=T, and the preceding will be reduced to 
{7^-{-p)v=X. 
Whence 
V={7^-\-p) ^X: 
where 
—pi \/ — I 
X- 
2p2 V — 1 
If v=v^-\-V 2 , we may evidently make 
{r—pis/-\)Vi = AX, {r-^piA/^\)v^=- 
AX; 
for these lead to the same result. And thus is found from two equations of a lower 
order than the given equation. 
If we now put for t its value in the two last equations, we shall have 
l'yi=AX, 1*^2= — AX' 
