Solving Partial Differential Equation

Question

Solving Partial Differential Equation $u_{xyy} (x,y,z) = 2 \sin x $

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Is this correct?

$$ \frac{\partial^3 u}{\partial y \, \partial y \, \partial x} (x,y,z) = 2 \sin x $$

Integrate respect to y

$$ \frac{\partial^2 u}{ \partial y \, \partial x} (x,y,z) = 2y \sin x + f(x) $$

Integrate respect to y

$$ \frac{\partial u}{ \partial x} (x,y,z) = y^2 \sin x + yf(x) + g(x) $$

Integrate respect to x $$ u(x,y,z) = -y^2 \cos x + yF(x) + G(x) + h(y) ;\quad \frac{\partial}{\partial x}F(x) = f(x),\quad \frac{\partial}{\partial x} G(x) = g(x) $$

Answer 1

$$ u_{xyy} = 2\sin x \\ u_{xy} = 2y\sin x + f(x,z)\\ u_x = y^2 \sin x + yf(x,z) + g(x,z) \\ u = -y^2 \cos x + yF(x,z) + G(x,z) + H(y,z) $$ where $F$ and $G$ are completely arbitrary functions ($F \in C^1$).