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# Class Note for MATH 250A with Professor Lega at UA

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Date Created: 02/06/15
Calculus and Differential Equations l MATH 250 A Differential equations of the form y gy gtllaramal muauonsonha form y g1 yl Calculus and gtllaramal Equatlonsl Existence dy i E g o Existence If g is continuous on the rectangle R tlyL lti fol S a lyiyol S b where a gt 0 and b gt 0 then there exists a continuously differentiable solution y of the above differential equation on lti tol g a for which yto YO Where b a min 37 7 M max M YElJuiblyu fbl lgyl This is a simplified version of the Cauchy Peano theorem gtllaramal equaluansoflhe form y flyl Calculus and gtllaramal Equatlonsl Uniqueness dy E g o Uniqueness If g is Lipschitz on R ie if there exists a constant kgt 0 such that for all y1 and y2 in the interval yo 7 b7 Yo b we have lgy1 7 gy2l S k lY1 7 y2l7 then there exists a unique solution y to the above differential equation on lti tol a such that yt0 yo The rectangle R and the number a are defined as in the previous theorem This is a simplified version of the Picard Lindelo39f Theorem gtllaramal muauonsonha form y g1 yl Calculus and gtllaramal Equatlonsl Examples 0 Conslder the dlfferentlal equatlon d l y2 Fora glven lnltlal condition yt0 yo is there always a solution that satisfies this initial condition 0 Yes 9 No 0 For the above differential equation is there a unique solution going through the point to7 yo 0 Yes 9 No d o For the differential equation is there a unique solution satisfying the condition y3 0 0 Yes 9 No gtllaramal equaluansoflhe form y g1 yl Calculus and gtllaramal Equatlonsl Equilibrium solutions dy dt y a A solution yX yo where yo is such that gyo 0 is an equilibrium solution of the above differential equation 9 To find all equilibrium solutions solve gy 0 for y 0 An equilibrium solution y yo is stable if there exists an interval I yo 7 5 yo e e gt 0 such that all solutions that start in I converge towards yo as t 4 00 0 An equilibrium solution is unstable if for every 5 gt 0 there always exists at least one solution starting in I Yo 7 6 yo e that moves away from yo as t 4 00 Method of solution dy E 7 Y dy 0 Re write the equation as 1 and Integrate both I I gy dt sides With respect to t 9 This gives an equation of the form Fy t C where C is an arbitrary constant and F is an antiderivative of 1g 9 Solution curves may therefore be obtained from one another by translation along the t axis 0 As before symmetries of g lead to symmetries of the family of solution curves Calculus and al Equations form v 91 quotl equations of If possible try to solve for y in order to obtain a family of explicit solutions Be very careful when solving for y since it is very easy to introduce functions that are not solutions Look for singular solutions which are solutions that are not part of the general solution given by Fy t C In particular make sure you have all equilibrium solutions If given an initial or boundary condition use it to find a particular solution Keep in mind existence and uniqueness theorems itial equations of the form y Calculus and Differential Equations Phase lines zgy y unstable d o Equilibrium solutions of d l gy are the values of y for which the graph of gy intersects the y axis o The stability of an equilibrium solution ye may be inferred from the sign of gy on each side of y ye Calculus and Dif Differential equations of the form yquot g1 Vl ntial Equations Bifurcation diagrams a When a parameter of the system is varied one says that a bifurcation occurs if new solutions appear or existing solutions change stability 0 A bifurcation diagram is a plot of all equilibrium solutions as functions of a parameter of the system Stable solutions are typically plotted with a solid stroke and unstable ones with a dashed stroke a The theory of dynamical systems provides a classification of bifurcations Classical examples are a The pitchfork bifurcation described by the differential dy 3 t 7 equa Ion dt by y d o The transcritical bifurcation described by 01 My 7 y2 d o The saddle node bifurcation described by 01 p 7 y2 mal equationsoflhe form y Calculus and D renual Equationsl

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