$1$ ODE as a boundary value problem $(50$ pts $)$

Consider a one$-$dimensional dimensionless heat conduction$-$convection problem

along a rod. The steady$-$state heat conduction equation with a nonlinear heat

source is given by:

T$($x$)$q$($T$($x$))=$T$($x$)$T$(0)=3\mathrm{.}0,$T$(1)=10\mathrm{.}0.$nn$=4,8,16,32($d$^(2)$T$($x$))/($dx$^(2))+$q$($T$($x$))=0$ for $0$

where T$($x$)$ is the dimensionless temperature distribution along the rod, and

q$($T$($x$))=$T$($x$)$ represents dimensionless heat convection. The boundary con$-$

ditions are:

T$(0)=3\mathrm{.}0,$T$(1)=10\mathrm{.}0.$

$(10$ pts$)$ Find its analytic solution.

$(20$ pts$)$ Use central difference to discretize the second order derivative

term and form the linear equations with three and n elements.

$(20$ pts$)($Coding$)$ Solve the linear equation $($refer to hw $1$ question $2,$ you

can reuse the code there or directly use MATLAB function.$)$ Plot and

discuss the solution with various number of elements, n$=4,8,16,32.$