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# Class Note for MATH 250A with Professor Lega at UA 2

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This 6 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 15 views.

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Date Created: 02/06/15
Calculus and Differential Equations l MATH 250 A Summary What differential equations are and how we st dy them 0 Examples of differential equations and of systems of differential equations a Identify the independent and dependent variables as well as the parameters ifany a Characteristic properties order linear vs nonlinear autonomous vs nonautonomous 0 Initial or boundary conditions are often given 0 Questions to be addressed in the study of a differential equation or a system of differential equations 9 Existence and uniqueness o Geometric considerations 0 Numerical solutions Analytical solutions Existence and uniqueness a Question Given an initial condition decide whether there exists a solution near the initial condition and if so whether it is unique 0 We have seen a variety of theorems that guarantee existence andor uniqueness of solutions to differential equations 0 For equations of the form y gX a For equations of the form y gy a For equations of the form y gx y For first order linear equations Geometric considerations 0 Decide where solution curves increase decrease or are concave up or down 0 For equations of the form y gx o For equations of the form y gy 0 You should be able to generalize the above to equations of the form V gX7y o Symmetries of the family of solution curves 5 given symmetries of the differential equation 5 o Iff is invariant under x A 7X then S is symmetric with respect to the yaxis o Iff is invariant under y A 7y then S is symmetric with respect to the X axis o Iff is invariant under x A 7X and y A 7y then Sis symmetric with respect to the origin Numerical solutions a For equations of the form y gx we have seen various ways of approximating integrals o For equations of the form y gxy we have discussed Euler39s method o In both cases we can sometimes decide whether an approximation is an underestimate or overestimate 9 We also discussed the various approximation errors associated with these methods 0 As part of the above we introduced Taylor polynomials as ways to approximate functions and discussed the resulting error Analytical solutions 0 For equations of the form y gx o For equations of the form y gy o For equations of the form y gxy a Separable equations 0 Equations with homogeneous coefficients 2 Linear equations 0 Bernoulli equations 0 Solving a differential equation always involves evaluating an integral We have seen various methods of integration Substitutions 0 Integration by parts Method of partial fractions 0 Trigonometric substitutions

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