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=Problem 7.1.1: Solve the Logistic Growth ODE Analytically=

Given: The Verhulst Model for Logistic Population Growth
The Verhulst Model, as it pertains to logistic population growth, describes the following system dynamics.
 * LHS: The time derivative $$ \displaystyle \dot{x}(t)= \frac{dx}{dt} $$, is the rate that the population changes with respect to time.
 * RHS: Is directly proportionate to the following:
 * $$ \displaystyle x(t):= $$ the population size at time $$ \displaystyle t$$.
 * $$ \displaystyle \left(1-\frac{x(t)}{x_{max}} \right):= $$ the remaining available resources.
 * $$ \displaystyle x_{max}:= $$ the carrying capacity which is maximum number of inhabitants for an isolated system.
 * $$ \displaystyle r:= $$ the constant of proportionality which serves as the rate constant. We are considering the case of population growth where $$ \displaystyle r>0 $$.

Find: An Analytical Solution for the Population Size at Time $$ t \in [0,20] $$

 * Obtain the analytical solution by solving the ODE shown in equation (7.1.1) for the following parameters.
 * Given the population carrying capacity $$ \displaystyle x_{max}=15 $$.
 * Given the population growth rate $$ \displaystyle r = 1.4 $$.
 * Given the initial condition $$ \displaystyle x_0 = 3 < \frac{1}{2}x_{max} $$.
 * Given the initial condition $$ \displaystyle x_0 = 9 > \frac{1}{2}x_{max} $$.

Step 1: Separation of Variables
Separating the independent and dependent variables in equation (7.1.1) results in the following expression that can then integrated.

Step 2: Partial Fraction Decomposition
Next we will consider the integrand and decompose its fractional form into two the sum of two elementary integrals.

Multiplying both the LHS and RHS by the denominator on the LHS results in the following, The final expression in (7.1.4) can be written in terms of its respective powers of $$ \displaystyle x $$, forming linear system of two equations with two unknowns. The unknowns we aim to obtain are $$ \displaystyle a $$ and $$ \displaystyle b $$. Since $$ \displaystyle a $$ is coefficient to $$ \displaystyle x^0=1 $$, we immediately obtain,

Next we observe the coefficients of order one $$ \displaystyle x $$ and using our solution from (7.1.5) $$ \displaystyle a=1 $$ to get,

Now that we have a solution to $$ \displaystyle a $$ and $$ \displaystyle b $$ we can substitute them into (7.1.3) to obtain an integrand that is integrable by elementary techniques.

Step 3: Direct Integration
Next we will simply evaluate the integral and perform the algebra necessary in obtaining and explicit expression for $$ \displaystyle x(t) $$ Which gives us the Verhulst Model for logistic population growth.

Step 4: Using the Data Set Parameters for $$ t \in [0,20] $$

 * Population carrying capacity $$ \displaystyle x_{max}=15 $$.
 * Population growth rate $$ \displaystyle r = 1.4 $$.
 * Initial condition $$ \displaystyle x_0^{(1)} = 3 < \frac{1}{2}x_{max} $$.


 * Population carrying capacity $$ \displaystyle x_{max}=15 $$.
 * Population growth rate $$ \displaystyle r = 1.4 $$.
 * Initial condition $$ \displaystyle x_0^{(2)} = 9 > \frac{1}{2}x_{max} $$.