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

An integrating factor directly solves a linear first-order equation; a general second-order equation needs an additional reduction. There are several sufficient cases, with all divisions restricted to intervals where their denominators are nonzero.

If $A=0$ and $B\ne 0$, divide by $B$ to obtain $y^{\prime }+(C/B)y=D/B$. The integrating factor is $\mu =\exp (\int C/Bdx)$ andIf $C=0$ and $A\ne 0$, put $u=y^{\prime }$. Then $u^{\prime }+(B/A)u=D/A$ is first order, with integrating factor $\nu =\exp (\int B/Adx)$; solve for $u$ and integrate once more for $y$.

A further reducible class has $C=B^{\prime }-A^{\prime \prime }$, for thenOne integration produces a first-order equation for $y$, whose integrating factor after division by $A$ is $\exp (\int (B-A^{\prime })/Adx)$.

Multiplication can always put a regular second-order equation into self-adjoint form by choosing $w=\exp (\int (B-A^{\prime })/Adx)$, sinceHowever, the remaining $wCy$ term generally prevents one integration from solving it. The limitations of integrating factors for second-order equations explain why this formal transformation alone is not a general solution method. These sufficient classes do not purport to classify every specially solvable coefficient choice.

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.