So I have this problem from the Elementary Differential Equations by Kells:
Solve this differential equation with the corresponding replacement:
$$
(st+1)t\cdot ds+(2st-1)s \cdot dt=0$$
with
$$z=st$$
So my question is whether am I doing something wrong since I did the exercise but I'm not getting the same result (which should be $(st-2)^3t=cs$).
What I did is to make the substitution, take the differentials of it ($dz=dx\cdot y+x\cdot dy$) and then to put it into the main equation, which leaves:
$$
(z+1)t ds + (2z-1)s dt = 0
$$
Having $dt=\frac{dz-ds\cdot t}{s}$,
$$
zt\cdot ds+t\cdot ds+(2zs-s)(\frac{dz}{s}-\frac{ds\cdot t}{s})=0
$$
which leads finally to:
$$
st+c=ln(st)
$$
What am I understanding wrong about the process?
Thanks in advance!