Solve and assess an exponential rod-expansion model

$dL/dt=kL$, with $k>0$ because the length is increasing.

$\int dL/L=\int k\,dt$ gives $\ln L=kt+C$ since $L>0$. Thus $L=e^Ce^{kt}=Ae^{kt}$, with $A=e^C>0$.

$L(0)=50$ gives $A=50$, so $L=50e^{kt}$ for $t\ge0$. The graph starts at $(0,50)$, is increasing and curves upwards.

For constant $k>0$, the model predicts unbounded, ever-faster growth as $t$ increases. The printed solution states that expansion cannot be non-terminating; this unlimited exponential prediction does not appropriately describe a heated metal rod over indefinitely long times.