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#.
