Solve yz dx + 2xz dy + xy dz = 0

Solve \[ yz\,dx + 2xz\,dy + xy\,dz = 0. \]

Step 1 · Check exactness and integrability

With $P=yz$, $Q=2xz$, $R=xy$: \[ P_y=z,\ Q_x=2z;\qquad P_z=y,\ R_x=y;\qquad Q_z=2x,\ R_y=x. \] Not exact, but \[ P(R_y-Q_z)+Q(P_z-R_x)+R(Q_x-P_y)=-xyz+2xz\cdot0+xyz\cdot1-\cdots=0, \] confirming an integrating factor exists.

Step 2 · Find $\mu=x^ay^bz^c$

Imposing all three exactness conditions on $\mu=x^ay^bz^c$ gives $b=2a+1$ and $c=a$; the simplest choice $a=c=0$, $b=1$ gives \[ \mu = y. \]

Step 3 · Multiply through and integrate

\[ y^2z\,dx+2xyz\,dy+xy^2\,dz=0, \] which is exact with potential $F=xy^2z$ (immediate: $F_x=y^2z$, $F_y=2xyz$, $F_z=xy^2$).

Verification

$d(xy^2z)=y^2z\,dx+2xyz\,dy+xy^2\,dz$; dividing by the integrating factor $y$ recovers $yz\,dx+2xz\,dy+xy\,dz=0$. \checkmark