Exponential growth

Example — Exponential growth: $y'=ky$

The simplest ODE: $\dfrac{dy}{dx} = ky$. We separate: $\dfrac{dy}{y} = k\,dx$. We integrate: $\ln|y| = kx + C$, hence $y = Ae^{kx}$ with $A = e^C$.

With initial condition $y(0) = y_0$ we get $A = y_0$, therefore: $\boxed{y(x) = y_0\,e^{kx}}$

The sign of $k$ decides the behaviour:

• if $k>0$: exponential growth (population, compound interest);
• if $k<0$: exponential decay (radioactivity).

Links

Topics: Differential equations
Concepts: Initial condition · Exponential growth · Decay · Separable variables
Functions: Exponential function
Skills: Integrating · Solving equations