= Solution
Differentiate the constraint to obtain $z'=2t-x'-y'$ and eliminate $z'$ from the other equations. Adding the resulting equations gives $y'=t-x+y/2$, and substitution then gives $x'+x=2t$. Its <integrating factor> is $e^t$, so $x=2t-2+C_1e^{-t}$; the initial value fixes $C_1=4$.
The remaining equation is $y'-y/2=-t+2-4e^{-t}$. Its integrating factor is $e^{-t/2}$, giving $y=2t+(8/3)e^{-t}+C_2e^{t/2}$; the initial value fixes $C_2=0$. Finally use the constraint:
$$
\boxed{\begin{aligned}
x(t)&=2t-2+4e^{-t},\\
y(t)&=2t+\frac83e^{-t},\\
z(t)&=t^2-4t+2-\frac{20}3e^{-t}.
\end{aligned}}
$$
The <acceleration> is $(4e^{-t},(8/3)e^{-t},2-(20/3)e^{-t})$, which is not constant. Thus \b[a constant net force cannot maintain this motion for a particle of constant mass], by <Newton's second law>. This conclusion concerns the net force in the stated inertial-coordinate interpretation; a time-dependent force can produce it.
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