Problem $2$: Edge Effects in Steady Conduction

Consider the steady state heat transfer problem depicted below. The dimensionless coordinates are $\backslash ($ x $\backslash )$ and $\backslash ($ y $\backslash )$ and the dimensionless temperature is $\backslash (\backslash$theta$($x$,$ y$)\backslash ).$ Constant temperatures are maintained on the top and bottom of a solid sheet that is indefinitely long. The temperature of the sheet is made nonuniform by an imposed variation along the boundary at $\backslash ($ y$=0\backslash ).$

$-$ Solve for the temperature profile. $($Hint: when determining your eigenfunctions, keep in mind that your eigenfunctions must make physical sense.$)$

$-$ Your solution should be an infinite series. Plot the solution $($using Matlab or other software$),$ by approximating the temperature profile using the first $7$ terms. Please solve it rightly with all steps as its from my exam! Thanks!