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Unformatted text preview: First Order Differential Equations; Existence and Uniqueness of Solutions Let us recall that a first order initial value problem consists of a differ- ential equation and specified initial condition given at some value x of the independent variable: dy dx = f ( x, y ) , y ( x ) = y . We have seen for several types of first order ODEs that we can find general solutions and corresponding solutions of initial value problems. However, there are many first order differential equations for which no solution in closed form can be found. In such cases the first thing we want to know is whether solutions exist or not and the second thing we want to know is whether there is only one solution for a given initial condition. For this reason the following theorem plays a central role. Theorem Let f ( x, y ) be defined in a region (open, connected set) R ⊂ R 2 with the properties: i) Let B be any bounded subset of R with the property that, if ( x, y 1 ) ∈ B and ( x, y 2 ) ∈ B , then all points ( x, y ) with y between y 1 and y 2 also lie in B . Then there is a positive constant L = L ( B ) such that, for all pairs ( x, y 1 ) and ( x, y 2 ) in B , | f ( x, y 1 )- f ( x, y 2 ) | ≤ L | y 1- y 2 | ; ii) The only possible discontinuities of f ( x, y ) in R are jump discontinuities across a discrete set of lines x = x k , k = 1 , 2 , 3 , ... . The x k have no accumulation points and f ( x, y ) has right and left hand limits along each line x = x k . Then for any ( x , y ) in R the initial value problem dy dx = f ( x, y ) , y ( x ) = y has a unique solution y ( x ) on some maximal interval ( α, β ) , α < x < β . 1 Remark The condition i) is satisfied if ∂f ∂y is continuous, or piecewise continuous, in R . For then it is bounded on any bounded set B , vextendsingle vextendsingle vextendsingle ∂f ∂y ( x ) vextendsingle vextendsingle vextendsingle ≤ C B , and using the mean value theorem we have, for ( x, y 1 ) , ( x, y 2 ) ∈ B and some η between y 1 and y 2 , | f ( x, y 1 )- f ( x, y 2 ) | = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ∂f ∂y ( x, η ) ( y 1- y 2 ) vextendsingle...
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- Spring '10