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Unformatted text preview: Math 334 Lecture #6 § 2.4: First Order ODE’s, Linear vs Nonlinear A central question for ODE’s is under what conditions does the IVP y = f ( t, y ) , y ( t ) = y , have a unique solution? [We have seen several examples where we have found a solution of an IVP, but had not argued that that was the only solution.] The Fundamental Existence and Uniqueness Theorem. If f and ∂f/∂y are continuous on an open rectangle α < t < β , γ < y < δ containing the point ( t , y ), then on some interval I = ( t- h, t + h ) contained in ( α, β ) for h > 0, there is a unique solution y = φ ( t ) of the IVP. [Sketch the picture of the open rectangle and the graph of the solution on I .] [NOTE: The conditions on f and ∂f/∂y guarantee that the solution φ ( t ) has a continuous derivative on I .] Example. For the nonlinear IVP y = y 2 , y ( t ) = y , the direction field f ( t, y ) = y 2 and its partial with respect to y , ∂f/∂y = 2 y , are contin- uous on the whole of the ty-plane. ! 0.4 t 0.0 ! 1.6 1 ! 4 ! 5 2.0 0.4 2 1.6 ! 1.2 1.2 ! 1 ! 3 5 ! 2.0 4 3 ! 2 0.8 ! 0.8 y(t) By the Fundamental Existence and Uniqueness Theorem, there is through each point ( t , y...
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
- Spring '11