Integral Problem

[Jason76]

I think this is a another "separation of variables" situation:

Let \(\displaystyle F\) be the number of trees in the forest at time \(\displaystyle t\), in years. If \(\displaystyle F\) is decreasing at a rate given by the equation \(\displaystyle \dfrac{dF}{dt} = -2F\) and if \(\displaystyle F(0) = 5000\), then \(\displaystyle F(t) =\)

Answer: \(\displaystyle 5000 e^{-2t}\)

 

[mathmari]

\(\displaystyle \dfrac{dF}{dt} = -2F => \dfrac{dF}{F} = -2dt \)
By integrating, we have:
\(\displaystyle ln(F)=-2t+c\)
\(\displaystyle e^{ln(F)}=e^{-2t+c} \)
\(\displaystyle F=e^{-2t}e^c \)
\(\displaystyle F=e^{-2t}c_{1} \)

To find the value of \(\displaystyle c_{1}\), \(\displaystyle F(0)=5000 => e^{0}c_{1}=5000 => c_{1}=5000 \)

So, \(\displaystyle F=5000e^{-2t} \)

 

[MarkFL]

I think this is a another "separation of variables" situation:...

Separation of variables is one way to go...there's also treating it as a linear equation, where we use an integrating factor, or we can superimpose the homogeneous solution and a particular solution, assume its form or use the annihilator method to derive it...fun for the whole family.