1. **State the problem:** Verify if the differential equation $$z \, dx + z \, dy + (x + y + \sin z) \, dz = 0$$ is exact and if so, find its primitive function. 2. **Recall the condition for exactness:** A differential equation of the form $$M(x,y,z) \, dx + N(x,y,z) \, dy + P(x,y,z) \, dz = 0$$ is exact if there exists a function $$F(x,y,z)$$ such that $$\frac{\partial F}{\partial x} = M$$, $$\frac{\partial F}{\partial y} = N$$, and $$\frac{\partial F}{\partial z} = P$$. 3. **Identify the functions:** Here, $$M = z$$, $$N = z$$, and $$P = x + y + \sin z$$. 4. **Check exactness conditions:** For exactness, the mixed partial derivatives must satisfy: $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}, \quad \frac{\partial M}{\partial z} = \frac{\partial P}{\partial x}, \quad \frac{\partial N}{\partial z} = \frac{\partial P}{\partial y}$$ Calculate each: - $$\frac{\partial M}{\partial y} = \frac{\partial z}{\partial y} = 0$$ - $$\frac{\partial N}{\partial x} = \frac{\partial z}{\partial x} = 0$$ - $$\frac{\partial M}{\partial z} = \frac{\partial z}{\partial z} = 1$$ - $$\frac{\partial P}{\partial x} = \frac{\partial (x + y + \sin z)}{\partial x} = 1$$ - $$\frac{\partial N}{\partial z} = \frac{\partial z}{\partial z} = 1$$ - $$\frac{\partial P}{\partial y} = \frac{\partial (x + y + \sin z)}{\partial y} = 1$$ Since all these equalities hold, the equation is exact. 5. **Find the primitive function $$F(x,y,z)$$:** Integrate $$M = z$$ with respect to $$x$$: $$F = \int z \, dx = xz + h(y,z)$$ Differentiate $$F$$ with respect to $$y$$ and set equal to $$N = z$$: $$\frac{\partial F}{\partial y} = \frac{\partial}{\partial y} (xz + h(y,z)) = \frac{\partial h}{\partial y} = z$$ So, $$\frac{\partial h}{\partial y} = z \implies h(y,z) = yz + k(z)$$ Now, differentiate $$F$$ with respect to $$z$$ and set equal to $$P = x + y + \sin z$$: $$\frac{\partial F}{\partial z} = \frac{\partial}{\partial z} (xz + yz + k(z)) = x + y + k'(z) = x + y + \sin z$$ Therefore, $$k'(z) = \sin z \implies k(z) = -\cos z + C$$ 6. **Write the final primitive function:** $$F(x,y,z) = xz + yz - \cos z + C$$ 7. **Conclusion:** The given differential equation is exact and its primitive function is $$F(x,y,z) = xz + yz - \cos z + C$$, so the implicit solution is: $$xz + yz - \cos z = \text{constant}$$