Problem 01
$(3x - 2y + 1)~dx + (3x - 2y + 3)~dy = 0$
Solution 01
$(3x - 2y + 1)~dx + (3x - 2y + 3)~dy = 0$
Let
$z = 3x - 2y$
$dz = 3~dx - 2~dy$
$dy = \frac{1}{2}(3~dx - dz)$
Thus,
$(z + 1)~dx + (z + 3)[ \, \frac{1}{2}(3~dx - dz) \, ] = 0$
$(z + 1)~dx + \frac{3}{2}(z + 3)~dx - \frac{1}{2}(z + 3)~dz = 0$
$[ \, (z + 1) + (\frac{3}{2}z + \frac{9}{2}) \, ]~dx - \frac{1}{2}(z + 3)~dz = 0$
$(\frac{5}{2}z + \frac{11}{2})~dx - \frac{1}{2}(z + 3)~dz = 0$