$24\mathrm{.}10$ The temperature distribution in a tapered conical cooling fin $($Fig$.$ P$24\mathrm{.}10)$ is described by the following differential equation, which has been nondimensionalized:

$\frac{d^{2}u}{d{x}^2}+(\frac{2}{x})(\frac{du}{dx}-px)=0$

where $u=$ temperature $),x=$ axial distance $(0x1),$ and $p$ is a nondimensional parameter that describes the heat transfer and geometry:

$p=\frac{hL}{k}\sqrt[\phantom{2}]{1+\frac{4}{2m^2}}$

where $h=$ a heat transfer coefficient, $k=$ thermal conductivity, $L=$ the length or height of the cone, and $m=$ the slope of the cone wall. The equation has the boundary conditions:

$)=(0)=(1$

Solve this equation for the temperature distribution using finite$-$difference methods. Use second$-$order accurate finitedifference formulas for the derivatives. Write a computer

FIGURE P$24\mathrm{.}10$