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

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

Calculus and Differential Equations l MATH 250 A Differential equations of the form y gx mai equations of the form i Calculus and D rential Equationsl Formal solutions 9 The differential equation y gx where g is continuous may formally be solved by integration so that yX gx dx C o Fundamental theorem of Calculus if f is a continuous function on 37b and if fX FX then b a X dx Fb a Fa 0 Second fundamental theorem of Calculus if f is continuous on 37b and ifx 6 37b then iax t dt fx al Equations Form al solutions continued 0 We can therefore write the general solution to y gx as yoeXgadrc o If we can find an antiderivative of gx then we have an explicit solution 0 Given a continuous function g can we always find an explicit expression for an antiderivative of g 0 Yes 9 No 0 There exists functions which we do not know how to integrate An example is the error function erfX 0X expit2 dt itial equations of the form iv Calculus and Differential Equations 0 Even if we do not know how to integrate g we may still be able to say something about the behavior of yx by looking at properties of the antiderivative of g 9 As X a 00 the integral in the expression for y becomes an 0 improper integral of the form gt dt 2 We will study improper integrals in the second semester of this course 0 An initial or boundary condition of the form yxo yo allows us to pick a particular solution from the one parameter family of solutions X ywngm X D o A solution curve is the graph of a particular solution yx Differential equations of the form y gm Calculus and Du ntial Equationsl Existence and uniqueness of solutions a TrueFalse All solution curves of y gx may be obtained by vertical translation of one of them 0 True 9 False 0 Since we assume that g is continuous we know that solutions exist Are they unique 0 If there is a unique solution for any initial condition is it possible for solution curves to cross or meet at a point 0 Yes 9 No a If at some point g becomes singular eg if it is undefined or it stops being continuous the argument for existence which assumes that g is continuous will fail 0 Nevertheless in some cases it will be possible to patch solutions of a differential equation found in adjacent intervals m Calculus and D ual Equauons Qualitative properties of solutions 0 Since we know y and assuming that g is smooth we know all of the derivatives of y a As a consequence we know when the solution y is increasing or decreasing and we know the concavity of the graph of y c We can use symmetries of the equation to relate a solution to another solution For instance if yx is a solution and gx is odd then ux yix is also a solution 0 If yx is a solution and gx is even then which of the solutions below is also a solution 9 quot0 yX 9 quot0 x 9 None of the above 0 Note that symmetries tell you about properties of the family of solutions not of each particular solution ghl Calculus and I a Equations Example of application 2 x 1 Consider the differential equation y 2 1 X 7 o Truefalse Solution curves increase for x in 711 0 True 9 False o Truefalse Solution curves are concave up for x 2 0 0 True 9 False o Truefalse Isoclines are all parallel to the y axis 0 True 9 False o Truefalse The family of solution curves is symmetric with respect to the y axis 0 True 9 False ual Emmaquot Spe fields 7 i A slope field for the differential equation y gx is a collection of line segments in the xy plane such that the slope of the segment centered at point xy is equal to gx 0 Given an initial condition yx0 yo one can sketch the associated solution curve assuming it exists and is unique by following the slope field starting from XOIYO a You should enter the slope fied program into your calculator In class we will use PPLANE to plot slope fields Slope field for the differential equation X2 1 y X2 7 1y plotted with the program PPLANE Calculus and I Ial equauon II the form y al Equauons Some of your questions Uniqueness 9 Why is it important 0 Can we see an example of a differential equation for which solutions are not unique 0 Are singularities in gx related to uniqueness issues 0 How is uniqueness related to non intersecting solution curves on ential amauonsofllie form V gm Calculus and Diif ntial Equationsl Some of your questions continued Solutions 0 Do differential equations always have an infinite number of solutions o If so do they always differ by a constant 0 How do we know when a differential equation cannot be solved 0 What happens to solutions of y gX as X goes to infinity 0 Can an explicit solution have a vertical tangent o If gx has a vertical tangent at x0 does it mean that if one solves dxdy 1gx one would have an horizontal tangent at that point o If we solve dydx gx and dxdy 1gx do we get the same solution curves Calculus and Differential Equations Differential amauonsofdie form y gm Some of your questions continued Slope fields 0 How far in X and y should we go o How far apart should the points in the x y plane be 0 What do isoclines mean Why are slope fields useful necessary How do we identify a singularity in a slope field If there exists isoclines of infinite slope can one construct a solution that spans both sides of the isocline Why can t we just use slope fields to solve ode s Calculus and Dilf ntial Equationsl on enlialamalionsofllieharmy gm

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