Find the general integral of the homogeneous equation…

Find the general integral of the homogeneous equation $\dfrac{ds}{dt}=\dfrac{s}{t}-\dfrac{t}{s}$.

Explanation of Method

Substitute $s=vt$ to reduce to a separable equation in $v$ and $t$; the result involves a logarithmic term from integrating $1/t$.

With $s=vt$, $\dfrac{ds}{dt}=v+t\dfrac{dv}{dt}$, and $\dfrac{s}{t}-\dfrac{t}{s}=v-\dfrac1v$, so

\[t\frac{dv}{dt}=v-\frac1v-v=-\frac1v \quad\Longrightarrow\quad v\,dv=-\frac{dt}{t}.\]

\[\frac{v^2}{2}=-\ln|t|+C \quad\Longrightarrow\quad v^2=K-2\ln|t|.\]

\[\frac{s^2}{t^2}=K-2\ln|t| \quad\Longrightarrow\quad \boxed{\,s^2+2t^2\ln|t|=Ct^2.}\]

Final answer:

\[\boxed{\,s^2+2t^2\ln|t|=Ct^2\ \ \Longleftrightarrow\ \ s^2=t^2(C-2\ln|t|).}\]