Page 191 - Foundations Of Differential Calculus

174 9. On Differential Equations
and by differentiation,
xdy ln x + ydx − ydx ln x dy
x 2 = y .
We conclude that
2 2
x dy − y dx
ln x = .
2
yx dy − y dx
When this equation is differentiated, with dx set constant, we have
2 2
dx x d y +2xdxdy − 2ydx dy
=
2
x yx dy − y dx
2 2 2 2
y dx − x dy yx d y + xdy − ydx dy
+ 2 ,
2
(yx dy − y dx)
or
2 2
3
2
2
2
dx y xdxd y − y x dx d y +3yx dx dy 2
= 2
2
x (yx dy − y dx)
2
2
3
2
2
3
2
−y xdxdy + y dx dy − 2xy dx dy − x dy 3
+ 2 ,
2
(yx dy − y dx)
which by reduction gives
3 2 2 2 2 2 2 2 2
y xdxd y − y x dx d y +3yx dx dy − 2xy dx dy
4
3 2 2 2 3 3 y dx 3
+3y dx dy − 2xy dx dy − x dy − =0,
x
or
2 2 2 2 2 2 2
y x (y − x) dx d y +3yx dx dy x dy + y dx − 2y x dx dy (dx + dy)
4 3 4 3
= x dy + y dx .
294. Exponential quantities are removed by differentiation in the same
way as logarithms. If the given equation is
Q
P = e ,
where P and Q are any functions of x and y, the equation can be trans-
formed into the logarithmic equation ln P = Q, whose differential is
dP
= Q, or dP = P dQ.
P
There is no real difficulty if the exponential quantities are more compli-
cated. In this case, if one differentiation is not sufficient, then two or more
differentiations will solve the problem.