Exponential growth or decay

Let us say we have a linear equation $\frac{dy}{dt}+ay=q(t)$. In that case $u(t)=\exp (at)$. Then

$$\begin{array}{rl}(e^{at}y{)}^{\prime }& =e^{at}q(t)\\ e^{at}y& =\int e^{at}q(t)\\ y& =e^{-at}\int e^{at}q(t)\end{array}$$

Now if $q(t)=b$ is constant

$$\begin{array}{rl}{e}^{at}y& =\int b{e}^{at}dt\\ e^{at}y& =\frac{b}{a}{e}^{at}+c\\ y& =\frac{b}{a}+c{e}^{-at}\end{array}$$

Now observe that

$$\underset{t\to \infty }{\lim }y=\frac{b}{a}$$

if $a>0$. And ${\lim }_{t\to \infty }y$ doesn’t exist if $a<0$.

So that, a function of the form

y(t) = d + ce^{at} $$ is called **growing exponentially** while the function of the form

y(t) = d + ce^{-at}

is called **decaying exponentially**. Note that in the original equation $\frac{dy}{dt} +ay = q(t)$, it is growing if $a<0$ and decaying if $a>0$.