Page 199 - Foundations Of Differential Calculus

182 9. On Differential Equations
This together with the proposed equation gives

2 2
ydx − xdy + adxdy =2ydx dy − 2xdxdy.
2 2
However, since dy = y dx/x , when the differentials are eliminated, we
obtain
y 4 ay 2 2y 3 2y 2
y − + = − ,
x 3 x 2 x 2 x
or
3 3 2 2
x − y + axy =2xy − 2x y.
2 2
Now, whether this is consistent with the differential y dx − x dy =0 be-
comes clear when it is differentiated; that is,

2 2 2 2
3x dx − 3y dy + ax dy + ay dx =2y dx +4xy dy − 2x dy − 4xy dx,
or
2
2
dy 3x + ay − 2y +4xy
= .
2
dx 3y − ax +4xy − 2x 2
But since
dy y 2
= ,
dx x 2
we have
4 3 2 4 3 2
3x +4x y + ax y =3y +4xy − axy ,

or
3
3
4
3y +4xy − 4x y − 3x 4 3 2 2 3
axy = =3y + xy − x y − 3x .
x + y
From the finite equation already obtained, we have
3 2 2 3
axy = y +2xy − 2x y − x ,
and when this is subtracted from the previous equation there remains
3 2 2 3
0=2y − xy + x y − 2x ,
2
2
which factors into 0 = y −x and 2y +xy +2x = 0. Of these, the equation
2
2
y = x can be consistent with dy = y dx/x , but it does not satisfy the finite
equation previously found. Unless we let a = 0, or unless both variables x
and y are set constant so that dx = 0 and dy = 0 and all of the differential
equations are satisfied, the given equation cannot hold.