Question

Determining Exponential Growth and Decay in Exercise, use the given information to write an exponential equation for y. Does the function represent exponential growth or exponential decay?

dy/dt = 2y, y = 10 when t = 0

Answer 1

`y=10e^(2t)`

Growth function

Explanation:

We are given that

`(dy)/(dt)=2y`

y=10 when t=0

Taking integration on both sides then we get

`\int (dy)/(y)=\int 2dt`

`lny=2dt+C`

Using formula

`\int(dx)/(x)=lnx,\int dx=x`

`y=e^(2t+C)`

`y=e^C\cdot e^(2t)=Ce^(2t)`

`ln x=y\implies x=e^y`

`e^C=Constant=C`

Substitute y=10 and t=0

`10=C`

Substitute the value of C

`y=10e^(2t)`

When t tends to infinity then

`\lim_(t\rightarrow \infty)=\lim_(t\rightarrow\infty)10e^(2t)=\infty`

Hence, the exponential function is growth function.