Solve and assess an exponential rod-expansion model

When a metal rod is heated, its length $L$ increases at a rate proportional to $L$.

(i) Write a differential equation for $dL/dt$ in terms of $L$.

(ii) Show that its general solution is $L=Ae^{kt}$.

(iii) The initial length is $50$ cm. Sketch $L$ against $t$.

(iv) Explain why this differential equation does not appropriately describe the expansion of the heated rod.