Page 186 - Foundations Of Differential Calculus

9. On Differential Equations 169
285. Once the value of dy has been found, by repeated differentiation we
2 3 4
find the value of d y, d y, d y, etc. Since these have no determined value
unless some first differential is made constant, for convenience, let dx be
constant, and for an illustration, let us consider this example:
3
3
y + x =3axy.
By differentiation we obtain
2 2
3y dy +3x dx =3ax dy +3ay dx,
so that
2
dy ay − x
= .
2
dx y − ax
Once more we take differentials with dx constant and obtain
2 2 2 2 2 2 2
d y −ay dy − a xdy +2x ydy − 2xy dx + a ydx + ax dx
= .
2
dx (y − ax) 2
When we substitute for dy the value already obtained,
2
ay dx − x dx
2
y − ax ,
and divide by dx,wehave
2
2
2
2
d y = ay − x 2 2x y − ay − a x
2
dx 2 (y − ax) 3
2 2 2
ax + a y − 2xy
+ 2 ,
2
(y − ax)
or
4
2
3
2 2
4
d y = 6ax y − 2x y − 2xy − 2a xy
2
dx 2 (y − ax) 3
3
2a xy
= − 3 ,
2
(y − ax)
4 4 2 2
since from the finite equation we have 2x y +2xy =6ax y . In this way,
using the finite equation, these values can be transformed into innumerable
different forms.
286. A differential equation can be expressed in an infinite number of
ways, since it can be combined with the finite equation. Thus, with the
preceding example we obtained the differential equation
2 2
y dy + x dx = ax dy + ay dx,