Solve (6x+yz) dx + (xz-2y) dy + (xy+2z) dz = 0

Solve \[ (6x+yz)\,dx + (xz-2y)\,dy + (xy+2z)\,dz = 0. \]

Explanation of Method

Check exactness directly, then integrate successively to build the potential.

Step 1 · Check exactness

$P_y=z=Q_x$, $P_z=y=R_x$, $Q_z=x=R_y$. \checkmark Exact.

Step 2 · Find the potential $F$

From $F_x=6x+yz$: $F=3x^2+xyz+g(y,z)$. From $F_y=xz-2y$: $g_y=-2y \Rightarrow g=-y^2+h(z)$. From $F_z=xy+2z$: $h'(z)=2z \Rightarrow h=z^2+C_0$. Hence \[ F = 3x^2+xyz-y^2+z^2. \]

Verification

$F_x=6x+yz=P$, $F_y=xz-2y=Q$, $F_z=xy+2z=R$. \checkmark

Result

The general integral is:

\[ \boxed{ 3x^2+xyz-y^2+z^2 = C } \]