Chapter 5.2, Problem 30E

Additional Boundary-Value Problems

Temperature In a Ring The temperature u(r) in the circular ring shown in Figure 5.2.11 is determined from the boundary-value problem

$r\frac{d^{2}u}{d{r}^2}+\frac{du}{dr}=0,u\left(a\right)=u_0,u\left(b\right)=u_1$,

FIGURE 5.2.11 Circular ring in Problem 30

where u₀ and u₁ are constants. Show that

$u\left(r\right)=\frac{u_0\ln \left(r/b\right)-u_1\ln \left(r/a\right)}{\ln \left(a/b\right)}$.