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Your reasoning is essentially correct, apart from the last paragraph. To conclude, note that Newton's equation: $$\ddot {\mathbf r}(t) = \mathbf f(\mathbf r (t)),$$ with initial condition $\mathbf x (0)=(x_0,0,0)$, $\dot {\mathbf x} (0)=(0,0,0)$ can be solved by puttin $y(t)=z(t)\equiv 0$, thus reducing to a one dimensional problem: $$\ddot x (t)=f(x(t)),$$ ...



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