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**Unformatted text preview: **2.4: DIFFERENCES BETWEEN LINEAR AND NONLINEAR EQUATIONS KIAM HEONG KWA We have so far considered a number of initial-value problems of the form (1) y = f ( t,y ) , y ( t ) = y , where the rate function f ( t,y ) is continuously differentiable (within its domain). Chances are we will be unable to solve (1) explicitly. This leads us to the following questions. (a) How do we know (1) actually has a solution if we cant exhibit the solution explicitly? (b) How do we know (1) has precisely one solution? (c) What is the use of determining whether (1) has a unique solution if it cant be exhibited explicitly? The answer to the third question lies in the observation that it is never necessary, in applications, to find the solution of (1) to more than a finite number of decimal places. This can be done numerically with the aid of a digital computer. Hence the knowledge that (1) has a solution and the solution is unique is our hunting license to go calculating it. Nonlinear Initial-Value Problems. Theorem 1. If the rate function f ( t,y ) and its derivative f y with respect to y are continuous in some rectangle R = { ( t,y ) || t- t | a, | y- y | b } on the ty-plane that contains the point ( t ,y ) , where < a < and < b < , then the initial value problem (1) is guaranteed to have a unique solution y ( t ) defined on the interval ( t- h,t + h ) , where h = min a, b M and M = max ( t,y ) R | f ( t,y ) | . Date : January 2, 2011. 1 2 KIAM HEONG KWA Note, however, that the existence and uniqueness of the solution is guaranteed only locally in the sense that the interval ( t- h,t + h ) is generally smaller than the interval [ t- a,t + a ] . Example 1 (Problem 11 in the text) . Consider the initial-value prob- lem (2) y = 1 + t 2 3 y- y 2 , y ( t ) = y ....

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