Differential equation word problem - Malthus's law Another problem I'm trying to solve that I have no idea what to do with. Here it goes:
In 1798, Rev. Robert Malthus proposed that the rate of change of a population is proportional to the actual population at any given time. 
If the world population was 3.712 billion in 1970 and 4.453 billion in 1980, then what does Malthus's law predict the world population to be in 2008?
 A: General edit.
If $p(t)$ is the population as a function of the time $t$ and $k$ is the constant of proportionality, the differential equation that models this law is
$$\dfrac{dp}{dt}=kp.\tag{1}$$
To find the solution $p(t)$ integrate $(1)$ and use the given values 
$$p(t_0)=p(1970)=3.712\, \mathrm{billion},\tag{2a}$$
$$p(1980)=4.453\, \mathrm{billion},\tag{2b}$$
where $t_0=1970$ is the initial instant. The equation $(1)$ can be integrated by the method of separation of variables, rewriting it as
$$
\begin{equation*}
\frac{dp}{p}=k\, dt\tag{3}
\end{equation*}
$$
and integrating both sides
$$
\begin{eqnarray*}
\int \frac{dp}{p} &=&\int k\, dt \\
\ln p &=&kt+C,\tag{4}
\end{eqnarray*}
$$
where $C$ is the constant of integration. The equation $(4)$ can be rewritten in different forms, such as
$$
\begin{eqnarray*}
p(t)&=&e^{kt+C}=Be^{kt}\tag{5a} \\
p(t) &=&Ae^{k(t-t_0)}=Ae^{k(t-1970)},\tag{5b}
\end{eqnarray*}
$$
where $A$ and $B$ are constants related to $C$. To proceed we could use either $(5\mathrm{a})$ or $(5\mathrm{b})$. Choosing equation $(5\mathrm{b})$, we find $A$ using the initial population $p(t_0)$ $(2\mathrm{a})$:
$$p(1970)=Ae^{k(1970-1970)}=Ae^0=A= 3.712\text{ billion}.\tag{5b'}$$ 
Now we can find the constant $k$ using the $1980$ population $(2\mathrm{b})$. We get the equation  $$4.453=3.712e^{k(1980-1970)},\tag{5b''}$$
whose solution, by applying logarithms, is 
$$k=\frac{1}{10}\ln\frac{4.453}{3.712}\approx 1.8201\times 10^{-2}\tag{5b'''}.$$ 
We thus have the following solution of $(1)$ that satisfies the initial conditions $(2\mathrm{a,b})$:
$$p(t)=3.712e^{1.8201\times 10^{-2}(t-1970)},\tag{6}$$
from which we obtain the Malthus's law prediction for the world population in $2008$:
$$p(2008)=3.712e^{1.8201\times 10^{-2}(2008-1970)}\approx 7.413\, \text{billion}.\tag{7}$$
Since $k>0$ the function  $(6)$ expresses an exponential growth.
