I am tasked to solve the equation in the figure for T(x) and then plot the temperature distribution through the sppon.

What is the temperature distribution plot of T(x) through the length of the spoon?

[Torsten]

I thought it was given by T(x) in the below formula. Isn't this the case ?

Answer 1

[Torsten]

The underlying differential equation is

lambda*V*d^2T/dx^2 = alpha*A*(T-T_inf)

with boundary conditions

T(0) = Tb, dT/dx(L) = 0

The solution is

T(x) = T_inf + (Tb-T_inf)*cosh(sqrt(alpha*A/(lambda*V))*(L-x))/cosh(sqrt(alpha*A/(lambda*V))*L)

So the m in your hint is

m = sqrt(alpha*A/(lambda*V))

This should suffice as hint how to plot the temperature profile over x.