**x** is the size of a population of bunnies. The rate of change of its growth is proportional to its size in time: $ \frac{d\mathbf{x}}{dt}=\dot{\mathbf{x}}=\lambda\mathbf{x} $ Separate variables and integrate: $ \frac{dx}{dt}=\lambda x\longrightarrow\frac{dx}{x}=\lambda\,dt $ $ \int\frac{dx}{x}=\int\lambda\,dt $ $ \ln x(t)=\lambda t+c $ $ x(t)=e^{\lambda t+c}=e^{\lambda t}K,\qquad K\, const $ How to find *K*? Use [[initial condition]], plug into $x(t)$: $ x(0)=e^0K=K $ Finally: $ x(t)=e^{\lambda t}x(0) $