Chapter 27, Problem 7P

Differential equations like the one solved in Prob. 27.6can often be simplified by linearizing their nonlinear terms. For example, a first-order Taylor series expansion can be used to linearize the quartic term in Eq. (P27.6.1) as

$\begin{array}{c}1\times {10}^{-7}{\left(T+273\right)}^4=1\times {10}^{-7}{\left(T_b+273\right)}^4+4\\ \times {10}^{-7}{\left(T_b+273\right)}^3\left(T-T_b\right)\end{array}$

where $T_b$ is a base temperature about which the term is linearized. Substitute this relationship into Eq. (P27.6.1), and then solve the resulting linear equation with the finite-difference approach. Employ $T_b=150\text{ and }\Delta x=0.01$ to obtain your solution.