Applications

Many real situations are modeled by the differential equation $y^{\prime }=ky$, where $k$ is a constant. Its solution is $y(t)=y_0{e}^{kt}$, with $y_0$ the initial value. If $k>0$, the quantity grows exponentially; if $k<0$, it decays exponentially. To use it, identify the rate of change, write $y^{\prime }=ky$, find $k$ from given data, and substitute the initial condition.