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Unformatted text preview: Differential Equations & Linear Algebra Lecture 3 Jianli XIE Email: xjl@sjtu.edu.cn Department of Mathematics, SJTU Fall 2010 Contents Modeling with first order ODE Contents Modeling with first order ODE Differences between linear and nonlinear equations Contents Modeling with first order ODE Differences between linear and nonlinear equations Equations with piecewise defined coefficients Contents Modeling with first order ODE Differences between linear and nonlinear equations Equations with piecewise defined coefficients Autonomous equations and population dynamics Exponential growth Logistic growth A critical threshold Logistic growth with a threshold Modeling with first order ODE Example At time t = 0 a tank contains Q lb of salt dissolved in 100 gal of water. Assume that water containing 1 4 lb of salt per gal is entering the tank at a rate of r gal/min and that the wellstirred mixture is draining from the tank at the same rate. Set up the initial value problem that describes this flow process. Find the amount of salt Q ( t ) in the tank at any time, and also find the limiting amount Q L that is present after a very long time. Modeling with first order ODE Solution The rate of change of salt in the tank, dQ/dt , is equal to the rate at which salt is flowing in minus the rate at which the salt is flowing out, that is, dQ dt = rate in rate out . The rate of change of salt is equal to the concentration times the flow rate. So rate in = r 4 , and rate out = Qr 100 . Therefore we obatin dQ dt = r 4 Qr 100 together with the initial condition Q (0) = Q . Modeling with first order ODE This is an initial value problem of a linear first order equation. We can find the general solution Q ( t ) = 25 + ce rt/ 100 . Using the initial condition, we get Q ( t ) = 25 + ( Q 25) e rt/ 100 . Taking the limit as t → ∞ , we have Q L = 25 . Existence and uniqueness for linear equations Theorem 1 If the functions p and g are continuous on an open interval I : α < t < β containing the point t = t , then there exists a unique function y = φ ( t ) that satisfies the differential equation y + p ( t ) y = g ( t ) for each t in I , and that also satisfies the initial condition y ( t ) = y , where y is an arbitrary prescribed initial value. Existence and uniqueness for nonlinear equations Theorem 2 Let the functions f and ∂f/∂y be continuous in some rectangle α < t < β, γ < y < δ containing the point ( t , y ) . Then in some interval t h < t < t + h contained in α < t < β , there is a unique solution y = φ ( t ) of the initial value problem y = f ( t, y ) , y ( t ) = y . Remark The conditions under which the solution exists uniquely are sufficient but not necessary. For linear equations, the solutions exist globally, while for nonlinear equations, the solutions exist locally. Example Example Discuss the existence and uniqueness of solutions of the initial value problem ty + 2 y = 4 t 2 , y (1) = y for different values of y ....
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 Fall '10
 JianliXie

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