Past exam of the mathematical methods I course of the University of Oxford / 2021 / 11 / b / Solution

Differentiate the constraint to obtain $z^{\prime }=2t-x^{\prime }-y^{\prime }$ and eliminate $z^{\prime }$ from the other equations. Adding the resulting equations gives $y^{\prime }=t-x+y/2$, and substitution then gives $x^{\prime }+x=2t$. Its integrating factor is $e^t$, so $x=2t-2+C_1{e}^{-t}$; the initial value fixes $C_1=4$.

The remaining equation is $y^{\prime }-y/2=-t+2-4e^{-t}$. Its integrating factor is $e^{-t/2}$, giving $y=2t+(8/3)e^{-t}+C_2{e}^{t/2}$; the initial value fixes $C_2=0$. Finally use the constraint:The acceleration is $(4e^{-t},(8/3)e^{-t},2-(20/3)e^{-t})$, which is not constant. Thus 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.