Chapter 7.4, Problem 71E

(a)

To determine

To show: The value of $L\left\{\ln t\right\}$ is $sL\left\{t\ln t\right\}-\frac{1}{s}$ using integration by parts..

(b)

To determine

The explicit solution $Y\left(s\right)$ of the foregoing differential equation and to show that solution of part(a) becomes $s\frac{dY}{ds}+Y=-\frac{1}{s}$ using theorem.

(c)

To determine

To show: The solution of part(b) becomes $L\left\{\ln t\right\}=-\frac{\gamma }{s}-\frac{\ln s}{s},s>0$ using the integral definition of Euler’s constant $\gamma =-{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-t}\ln tdt}$, where $\gamma =0.5772156649\dots$ and $Y\left(1\right)=-\gamma$.