On the other hand initial value problems will always have a unique solution - for initial value problems we do have the existence and uniqueness of a solution. Nonhomogeneous linear differential equations We will study nonhomogeneous second-order linear differential equations of the form ay 00 + by 0 + cy = G ( x ) where a, b, c are constants 36 CHAPTER 3. INITIAL VALUE PROBLEMS and exponentiating c k c = e( )teC in which eC is simply another constant. A particular solution is found by evaluating the constant for an initial value c= c 0 at t= 0 c(t) = c 0ke( )t k+ c 0 e( )t 1 The logistic model has our population c living in isolation, in a petri dish or on a deserted island perhaps. 9 Initial Value Problems, continued I Thus, part of given problem data is requirement that y(t 0) = y 0, which determines unique solution to ODE I Because of interpretation of independent variable t as time, we think of t 0 as initial time and y 0 as initial value I Hence, this is termed initial value problem, or IVP I ODE governs evolution of system in time from its initial state y View Notes - Lecture 1.2 - Initial-value problems.pdf from MATH 3260 at Georgia State University. MATH-3260: Diff. Eq. 1.2. 1.2. Initial-Value Problems Initial-Value Problems Lets review some. Study Resources. Main Menu; by School; by Literature Title; by Subject; by Study Guides; Textbook Solutions Expert Tutors Earn. - Initial value problems In this chapter we develop algorithms for solving systems of linear and nonlinear ordinary di erential equations of the initial value type. Such models arise in describing lumped parameter, dynamic models. Entire books (Lapidus & Seinfeld, 1971; Lambert, 1973) are devoted to the de-velopment of algorithms for such problems. Solving Initial Value Problems Jake Blanchard University of Wisconsin - Madison Spring 2008 Example Problem Consider an 80 kg paratrooper falling from 600 meters. The trooper is accelerated by gravity, but decelerated by drag on the parachute This problem is from Cleve Moler'sbook called Numerical Computing with Matlab (my favorite Matlab book) Theorem 1' says that all initial value problems have a solution, and no more than one solution. If we can show that we can satisfy every initial value problem at some pointt02 I, then we have all solutions. Is this clear? If not, why not? Should some sentences be read again? In our intervalI= (¡1;1), the easiest point to work with ist0= 0. By Theorem 2, we know the initial-value problem has a unique solution. Solve the initial value problem: dy dt 1 tsin yt , y 0 0. We cannot solve it exactly. Here is a numerical solution for this initial-value problem: 0 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 the graph of y t where y′ 1 sin yt ,0≤t ≤2, y 0 4. INITIAL VALUE PROBLEMS Notice that the error growth follows the di erence equation in (10), not the dif- ferential equation. The sti example with a t = ( 20)(1 9 ) gave a large 1 + a t, even when ea twas small. We still call forward Euler a stable method, because as soon as t is small enough the danger has passed. Initial-ValueProblems for Ordinary Differential Equations y(x) (X3'U3) I I I SLOPE=f(x2,u2) I I I I I I Define C~=feed concentration of benzene (mole/cm3) z=axial reactor coordinate (cm) L reactor length y dimensionless concentration of benzene (CB/C~) x=dimensionless axial coordinate (z/L). Initial-Value Problems for Ordinary Differential Systems 1.1. Introduction. We will study initial-value pr