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# Introduction to Differential Equations with Linear Algebra APPM 2360

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This 2 page Class Notes was uploaded by Dr. Filomena Hegmann on Thursday October 29, 2015. The Class Notes belongs to APPM 2360 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/231878/appm-2360-university-of-colorado-at-boulder in Applied Math at University of Colorado at Boulder.

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Date Created: 10/29/15

APPM 2360 Exam 1 review 1 0 Basics 7 An ODE is an equation involving y t and the various derivatives dky W up to some order n 7 y is the dependent variable t is the independent variable 71 is the order of the ODE 7 A solution is any function yt that works when plugged into the ODE 7 An IV is an ODE along with an initial condition that speci es the value of y at some particular value of t 0 Solutions For rst order equations as long as the rate function is nice ft y is continuous in some region then solutions exist Solutions to an ODE are not unique until an initial condition is speci ed If is continuous in a region around a given initial condition then the solution to the IV is unique The existence and uniqueness of this solution is guaranteed only locally however The general solution to an ODE is one that can satisfy any given initial condition A particular solution is one that satis es only one initial condition 239e it is one out of the in nitely many solutions contained in the general solution 0 Threepronged attack 7 Analytic try to solve the equation directly Gives exact function as answer but may not be possible for many equations 7 Qualitative try to determine general properties of solutions without actually solving Works on pretty much any equation and doesn t require hard math but doesn t give any exact values or even approximate values just general properties 7 Numerical try to get approximate values of the solutions Gives actual values and works on most equations but gives solution values only at certain values of t for particular solutions 239e requires an initial condition Gives no conceptual understanding Requires a computer in reality 0 Analytic 7 An equation is separable if it can be written as y gthy for some functions 9 h Can be solved by separation of variables 9 rewrite as ll 527 gt integrate with respect to t f dt fgtdt which is equivalent to f dy fgtdt evaluate the integrals and if possible solve for make sure your general solution actually includes equilibrium solutions that might have been lost by dividing by APPM 2360 Exam 1 review 2 7 An equation is linear if it can be written as y pty qt or more generally anty a2ty a1ty a0ty ft A linear equation is homogeneous if there is no function oft on its own 7 Le qt E 0 or ft E 0 Solutions to linear equations have special properties Given any number of solutions y1 yk to the homogeneous part of the equation the function yt clyl ckyk is also a solution to the homogeneous part for any set of constants 9 Given a solution 34 to the homogeneous part of the equation and a solution yp to the full equation the function yt y yp is also a solution to the full equation Moreover if y is the general homogeneous solution then y y yp is the general solution to the full problem Therefore one way to solve rst order linear equations is the Euler Lagrange Variation of Parameters approach Find the general solution 34 of the homogeneous part by separation of variables or by inspection Assume a form of the particular solution yp in which the constant C in y is allowed to be a function Ct Substitute this choice of yp into the equation to get an equation for C t Solve this equation to get C25 and hence yp 9 9 9 By linearity the general solution is y y yp However another approach is to use lntegrating Factors For the equation 3 pty qt de ne the integrating factor 1 5f WW Multiply the equation by ut The left hand side can now be rewritten as uty check back to make sure lntegrate both sides with respect to 25 don t forget the constant Solve for yt o Qualitative evaluating ft y at various points in the ty plane gives the slopes of the solutions of y fty that happen to pass through those points Drawing small lines with the appropriate slopes at a number of points creates a slope eld which gives a Visual idea of how the solutions behave 0 Numerical you have seen the simplest numerical method known as Euler s Method for obtaining approximate solutions to an IV at some given time T 7 Choose a stepsize h At the 0th step the start you have to and yo given by the initial condition For n 1 2 N where N Th 7 the smaller the stepsize the more steps you have to take repeat the following procedure 7 tn tn71 l h 7 39 yn71 l hftn717yn71 7 ln words the time is advanced in equal steps the solution is advanced by evaluating the slope where you currently are and stepping forward from the current value a distance h with that slope 7 Euler s Method is 9h or rst order77 h hl meaning that there error in the approximation is proportional to h In general numerical methods are classi ed as pth order 90117 if their error is proportional to 71 Generally higher order is better

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