Solution of Ex:3.1 Differential Equations. Differential equations with boundary-value problems are used in a variety of fields, including physics, engineering, and economics. One common application is modeling growth and decay, where a quantity changes over time according to some rate of change. In the 7th edition book, “Differential Equations with Boundary-Value Problems,” one such example is discussed in Ex:26, which deals with the growth and decay of a population.
Solved Exercise Solution of Ex:3.1 Differential Equations
Example for understanding
The Ex:26 differential equation with boundary-value problems is given as:
dP/dt = kP(M – P)
where P is the population, t is time, k is the growth rate constant, and M is the maximum carrying capacity of the population. The boundary conditions are P(0) = P0 and P(T) = PT, where P0 is the initial population and PT is the population at time T.
To solve this differential equation, we can use the method of separation of variables. We first rewrite the equation as:
dP/(P(M – P)) = kdt
Integrating both sides, we get:
ln|P/(M – P)| = kt + C
where C is the constant of integration. We can simplify this expression by using the identity:
ln|P/(M – P)| = ln|P| – ln|M – P|
Substituting back, we get:
ln|P| – ln|M – P| = kt + C
Taking the exponential of both sides, we get:
|P|/|M – P| = e^(kt+C) = De^kt
where D is the constant of integration. Since we are dealing with populations, we can assume that P and M – P are both positive, so we can drop the absolute values. Solving for P, we get:
P = MDe^(kt)/(1 + De^(kt))
Using the boundary conditions, we can solve for the constants D and C. We get:
D = (P0 – M)/(P0 – M + Me^(-kT))
and
C = ln|P0 – M| – ln|P0 – M + Me^(-kT)| – kT
Conclusion
Using these values, we can find the population at any time t. This differential equation with boundary-value problems demonstrates how we can use differential equations to model growth and decay, which is a common phenomenon in many fields. By solving the differential equation using separation of variables and using the boundary conditions to find the constants, we can obtain a solution that describes the population at any time t.
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