PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 

=-=( e 4 (=D te a+ = Be. ) 
[nm \q* a” ag ** aes 
pnt seal be n n (n+ 1) 
i ie ar a gets &e. ) 
yp R(_1 th DOD, y, 
ar m \aat? gat? ag ts ey 




x“ n 

dx 
ees Bae ee eee 4782 
=a’ e a6 da+/— 72 e 
n 
Pa 

Weep ee eee et ( 
y ée be jaf x dx+V/—1 iB ay 2 | 
7. The solution of the foregoing examples might have been obtained very 
differently, thus : 
by 3 
if d} y-a' y=X; y= {= Gt8e 

Now ae X is the solution of the ordinary differential equation ce —av=X; its 
value is, consequently, e*” ( a8 e* Xdar+ c) . Hence 
4 
(=z, 
===. et ( sof oe Xdax+ ) +a® e* (eo °* Xda + c) 
dx 
For instance, if X=0, the solution of the equation is 
y=2a* Ce%* ; 
which is the same as that given above. 
Se Hix: 3. ae ad'y 

This may be written (d+ad}+b).y=0; or (d}—a®) (d?—@*).y=0; where 
a} + B!=—a, and (a 8)?=8, or a}, GB are the roots of the equation 2? +az+6=0. 
: y=A(d}—at)-1.0+ Bd}—6?)-1.0 
=Ae**+Be®? (Ex. 1.) 
Cor. 1. If a=, we must write a+e instead of G, and proceed as in similar 
cases. 
The result is y=Ae** +Bae** 
Cor. 2. In precisely the same way we may find the solution of the equation 

