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Unformatted text preview: MA366 Sathaye Chapter 2 Summary Continued 1. Modeling. In section 2.3, we find the discussion of modeling - creating and solving mathematical equations which describe and/or approximate events in the real world. All work begins with carefully naming variables and the using known scientific laws to develop an equation. The equation is then solved subject to the given initial conditions. Effects of variation of the initial condition, nature of solution as the time approaches certain times or even infinity as well as the effect of change of the used parameters is often discussed. There is no single method to solve these problems. A firm graip on the solution techniques and a clearheaded understanding of the descriptive words is essential and lots of practice is the only insurance! 2. 2.4. Nature of solutions. We take stock of what is learned. • Linear equation Theorem 2.4.1 y + py = q can be solved when p,q are continuous and leads to a unique solution for given initial conditions. More importantly, the solution has the form f 1 ( t ) + cf 2 ( t ) where f 1 depends only on the LHS y + py , while f 2 depends on the RHS as well. The solutions are guaranteed throughout the region of continuity. • Non linear equation Theorem 2.4.2 y = f ( t,y ). The existence and uniqueness are guaranteed when f,f y are both continuous. The solution itself may not extend over the whole region of continuity but may be valid only in a restricted interval near the initial value t = t . The general solution has an arbitrary constant, but the solution does not have the neat form of explicitly solved y . Often, it may be impossible to give a formula for the solution of y in terms of t . Often only a relation between y,t is established. 3. Autonomous system 2.5. The equation has the form y = f ( y ) where f ( y ) is free of t . Thus it is a special type of separable equation. The existence theorems guarantee solution where f and f y are continuously differentiable. The equation often shows up in many practical problems and hence is extensively studied and analyzed....
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- Fall '09
- Equations, Constant of integration, Order theory, Lipschitz continuity, Picard–Lindelöf theorem