If we rewrite the given Diff. Eqn. (DE) as,
y'=dy/dt=y/t{1+ln(y/t)}, we immediately recognise it as a
Homogeneous DE of First order & First degree.
Its General Solution (GS) is obtained by using the subst.
y=tx," so that, "dy/dt=tdx/dt+x, and, y/t=x," as well".
Thus, the DE becomes,
tdx/dt+x=x{1+lnx}=x+xlnx, i.e., tdx/dt=xlnx,
dx/(xlnx)=dt/t"......[separable variable]".
:. {1/xdx}/lnx=dt/t.
Integrating, int(1/x)/lnxdx=int{d/dx(lnx)}/lnxdx=intdt/t+lnc.
:. ln(lnx)=lnt+lnc=ln(tc).
:. lnx=tc, or, x=e^(tc), i.e., y/t=e^(tc).
rArr y=te^(tc), is the desired GS!
"N.B. : The GS can be written as, "y=t(e^c)^t=tk^t, k=e^c.
