Page 188 - Foundations Of Differential Calculus

9. On Differential Equations 171
2 2
two equations y = ax + ab and y = ax are entirely different, since in
the first any constant quantity can be given to b. Nevertheless, when both
equations are differentiated, we obtain the same differential equation
2ydy = a dx.
2
Indeed, all of the equations represented by y = ax, depending on the value
assigned to a, correspond to a single differential equation that contains no
2
a. If this equation is divided by x, so that y /x = a, then this, when
differentiated, gives

2xdy − ydx =0.

Even both transcendental and algebraic equations can lead to the same
differential equation, as is seen in the equations
2 2 2 x/a
y − ax = 0 and y − ax = b e .

If each of these equations is divided by e x/a , so that we have
2
2
2

e −x/a y − ax = 0 and e −x/a y − ax = b ,
then when each of these is differentiated, the same differential equation
results:
2
y dx
2ydy − adx − + xdx =0.
a
289. The reason for this diversity is the fact that the differential of a
constant quantity is equal to zero. Hence, if a finite equation can be reduced
to such a form that some constant quantity stands alone, neither multiplied
nor divided by variables, then by differentiation there arises an equation
in which that constant quantity is completely eliminated. In this way any
constant quantity involved in a finite equation can be eliminated through
differentiation. Thus, if the given equation is
3 3
x + y =3axy,
and if this is divided by xy, so that we have
3 3
x + y
=3a,
xy
then when this equation is differentiated we have
3 3 4 4
2x ydx +2xy dy − x dy − y dx =0,
in which the constant no longer appears.