Solve #ty'=y(1+lny-lnt)# ?

1 Answer
Apr 13, 2018

#"The GS is, "y=te^(tc)#.

#"N.B. : The GS can be written as, "y=t(e^c)^t=tk^t, k=e^c#.

Explanation:

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#.