The simplest interesting autonomous equation says the rate is proportional to the amount. Separating it takes two lines and gives the exponential, and the sign of the constant decides everything: growth without bound one way, decay to nothing the other. The model is exact for radioactivity, good for money, and wrong for populations over any long stretch — which is what the next lesson repairs.

Interest paid on interest. Each unit of the balance earns at the same rate, so the balance earns in proportion to itself, and the total accelerates without anything about the rate ever changing.

Both curves solve $y'=ky$. A positive constant gives growth that steepens for ever; a negative one gives decay that flattens towards zero and never arrives.

The equation $y'=ky$ with $y(0)=y_{0}$ has the single solution . It has one equilibrium, , unstable when and stable when . The constant is a rate per unit of the input, and it is fixed by one measurement beyond the starting value.

• Half-life. Decay halves in a fixed time whatever the starting amount: $e^{kt}=\tfrac12$ gives $t=\frac{\ln }{}$.

Separating the proportional-rate equation gives the exponential, and the starting value is the factor in front.

The equation with a constant added is the same equation about a different level: shift the unknown by the equilibrium and the constant disappears.

Great work! You now know how to:

• solve the proportional-rate equation and use its solution
• compute a half-life or a doubling time, and say why the starting value drops out
• turn an equation with a constant term into this one by shifting
• say where the exponential model stops being honest