By Brannan J., Boyce W.
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Extra resources for Differential equations. An introduction to modern methods and applications
This corresponds to the choice c = 0 in the general solution (17). Solutions of the more general equation (15), in which the growth rate and the predation rate are unspecified, behave very much like those of Eq. (16). The equilibrium solution of Eq. (15) is p = a/r. Solutions above the equilibrium solution increase, while those below it decrease. 5 p − 450. CONFIRMING PAGES 17:38 P1: KUF/OVY JWCL373-01 P2: OSO/OVY QC: SCF/OVY T1: SCF JWCL373-Brannan-v1 10 Chapter 1 October 12, 2010 Introduction In each of the above examples, we note that the equilibrium solution separates increasing from decreasing solutions.
The purpose of Example 1 is to show you the details of implementing a few steps of Euler’s method so that it will be clear exactly what computations are being executed. Of course, computations such as these are usually done on a computer. Some software packages include code for the Euler method, whereas others do not. 1. The outline of such a program is given below; the specific instructions can be written in any high-level programming language. 3 Numerical Approximations: Euler’s Method 31 The Euler Method Step 1.
38. Consider the initial value problem y + ay = g(t), Assume that a is a positive constant and that g(t) → g0 as t → ∞. Show that y(t) → g0 /a as t → ∞. Construct an example with a nonconstant g(t) that illustrates this result. 39. Variation of Parameters. Consider the following method of solving the general linear equation of first order: y + p(t)y = g(t). y(0) = y0 3 2 y = 3t + 2et , y = A exp − y(0) = y0 . Find the value of y0 that separates solutions that grow positively as t → ∞ from those that grow negatively.
Differential equations. An introduction to modern methods and applications by Brannan J., Boyce W.