Math 2255 Fall 2020 Homework 2 Due September 11

Math 2255 Fall 2020homework 2due Friday September 1130ampleas

Determine (without solving the problems) an interval in which the solution to each initial value problem is certain to exist. Then, for nonlinear ODEs, identify where the hypotheses of the existence and uniqueness theorem are satisfied in the ty-plane, and describe the initial conditions that will guarantee a unique solution nearby.

Solve the IVP y' + 2y = g(t), y(0) = 0, where g(t) = 1 for 0 ≤ t ≤ 1 and g(t) = 0 for t > 1.

Describe the effect on the rabbit population P(t) with birth rate bP and no death. Consider possible stable or unstable equilibrium points, and draw phase lines with solution graphs.

Identify critical points and classify their stability for the differential equations:

• dy/dt = –k(y – 1)^2, with k > 0
• dy/dt = ay – b√y, with a > 0, b > 0
• dy/dt = y^2(1 – y)^2

Paper For Above instruction

This paper addresses key concepts in differential equations, especially those related to existence, uniqueness, stability, and qualitative analysis of solutions. The initial value problems (IVPs) are examined in terms of their domains of solution existence, based on the conditions set by the existence and uniqueness theorem. The nonlinear ODEs are analyzed by identifying regions in the ty-plane where the hypotheses hold and discussing the implications for initial conditions that ensure a unique and well-defined solution.

Starting with the initial value problem y' + 2y = g(t), y(0) = 0, where g(t) is piecewise defined, one observes that the nonhomogeneous linear differential equation can be approached via integrating factors. Since the coefficient of y is continuous everywhere, the domain of existence of solutions is all real t, but discontinuities in g(t) at t=1 imply potential changes in the solution's behavior at that point. The solution involves integrating g(t) with the integrating factor e^{2t} and accounting for the piecewise definition, leading to a solution that changes form at t=1. The integrability and continuity of g(t) on each subinterval guaranty the existence and uniqueness of the solution within each interval, according to standard theorems.

The rabbit population model with a birth rate proportional to P, accompanied by no death, reflects exponential population growth. Mathematically, P' = bP, which is a straightforward first-order linear ODE. The solution P(t) = P(0) e^{bt} illustrates exponential growth, indicating that, without resource constraints or mortality, the population will grow without bound. In terms of stability and equilibria, the only equilibrium point is at P=0, which is unstable, since any small positive initial population leads to unbounded growth.

Critical points in differential equations often correspond to equilibria where the derivative is zero. For dy/dt = –k(y – 1)^2, the critical point y=1 is found by setting the derivative to zero, and its stability is classified based on the behavior of solutions near y=1. Since the derivative is negative for y > 1 and positive for y

In the case of dy/dt = ay – b√y, with a > 0 and b > 0, the critical points are at y=0 and y = (b/a)^2. The zero solution is semi-stable, attracting solutions from above but not from below, while the positive equilibrium at y=(b/a)^2 is stable. These points are determined by setting the derivative to zero and analyzing the sign changes of dy/dt near each equilibrium, supplemented by phase line diagrams.

Finally, for dy/dt = y^2(1 – y)^2, the critical points occur at y=0 and y=1. The stability of these points depends on the behavior of solutions approaching or receding from these points, with y=0 being unstable and y=1) being stable. The phase line diagram illustrates the regions of attraction and repulsion, and solution graphs near these points show the long-term tendencies of the system.

References

• Boyce, W. E., & DiPrima, R. C. (2017). Elementary Differential Equations and Boundary Value Problems (11th ed.). Wiley.
• Chapra, S. C., & Canale, R. P. (2010). Numerical Methods for Engineers (6th ed.). McGraw-Hill.
• Hartman, P. (2002). Ordinary Differential Equations. SIAM.
• James, G., et al. (2006). Calculus: Early Transcendental (2nd Ed.). Pearson.
• Krantz, S. G. (1999). The Implicit Function Theorem: History, Theory, and Applications. Springer.
• La Salle, J., & Lefschetz, S. (1961). Stability by Liapunov's Direct Method. Academic Press.
• Perko, L. (2001). Differential Equations and Dynamical Systems (3rd ed.). Springer.
• Roberts, M. (2009). Mathematical Modeling of Biological Populations. CRC Press.
• Strogatz, S. H. (2014). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press.
• Zill, D. G. (2018). Differential Equations with Applications and Historical Notes (11th ed.). Cengage Learning.