LIN ALG & DIFF EQNS
LIN ALG & DIFF EQNS MATH 054
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This 2 page Class Notes was uploaded by Kavon Feest on Thursday October 22, 2015. The Class Notes belongs to MATH 054 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/226610/math-054-university-of-california-berkeley in Mathematics (M) at University of California - Berkeley.
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Date Created: 10/22/15
MATH 54 32153205 DIFFERENTIAL EQUATIONS SO FAR This is an outline of the major types of differential equation problems you should be able to solve now and potentially on Wednesdays quiz Feel free to skip those with which you7re most comfortable a good target is probably nishing 678 of these ADMINISTRATIVE STUFF I apologize for missing my sections on Friday for missing any appointments I had on that day and for giving no warning This was accidental I will have extra of ce hours in 848 Evans between 1 and 5 PM today Monday and between 9 and 1130 AM tomorrow Tuesday I may have lied about Abel s formula IfI wrote down any formula for Wy1y2 given two solutions to y pty qty 0 other than Wt Cexp 7 fptdt you should forget the erroneous formula It is stated correctly in Boyce amp DiPrima 1 TECHNIQUES AND SAMPLE PROBLEMS Given an easy linear homogeneous ODE nd a fundamental set of solutions So far the equations we should be able to solve are those with constant coe cients for now we only fully know how to proceed when the characteiistic polynomial has distinct real roots But you are expected to remember separable ODE7s from Math 13 Warning when writing down the characteristic polynomial don7t forget that y yw translates to 7 0 1 Exercise Do this for y3 7 y 0 Given a fundamental set and suitable initial conditions usually the values of ya y 1a nd the hopefully unique solution Exercise Do this for the ODE from the previous exercise with y0 12 yO 1 y 0 3 Given a set of functions compute their Wronskian Exercise Find W1 7 t1 t Given a wild type linear homogeneous ODE be able to write it in the standard form y p1ty 1 pn1ty 107 ty 0 for the purpose of doing the next two types of problems Exercise do this for the associated Legendre equation encountered during a quantum mechanical treatment of a hydrogen atom for instance 1 7 12M EM 71 7 y 0 71 lt z lt 1 Given an equation in this standard form determine the longest intervals on which you can expect a unique solution also determine the Wronskian Wyl yn of any set of n solutions to the nth order equation Which interval is longest might depend on your initial conditions which specify a value of z where your solution should be de ned Exercise do this for the ODE from the previous exercise Use Abel s formula when applicable to show a function or set of functions can t be a solution or set of solutions to a nice linear homogeneous ODE Exercise Prove that tsint cannot be a solution to any ODE y pty qty 0 in which pt qt are continuous Hint this function satis es fO 0 Also prove that y1t 1 and y2t cost cannot both be solutions to such an ODE Hint again consider t 0 An easier but less general method is to think of it as the solution of an initial value problem important Be able to translate an nthorder linear ODE and its initial conditions to a system rstorder linear ODE s Pretty much everything mentioned above has an analogue for such systems these analogues often look more natural than their forms for higherorder ODE7sl Exercise Do this for y 7 2y y 0 with initial conditions y0 0 y 0 1 Finally several techniques which are more closely related to linear algebra First be able to recognize whether an ODE or a differential operator is linear 1 Exercise ls y 7 ty t ly 0 a linear ODE ls Ly ny y y t a linear differential operator Testing linear independence of functions With and Without Wronskians Exercise If a family of functions is linearly dependent is its Wronskian zero If a family of functions has a Wronskian of zero must it be linearly dependent Explain Why y1t H and y2t ltl are linearly independent When considered as functions de ne on R 10 Applying ideas of basis and coordinates to more abstract problems Past homework exercise if yhyg is a linearly independent set of functions under What condi tions is alyl agyg blyl bgyg also a linearly independent set of functions Your answer should be When the set of coef cient vectors Z 11 Finding linear dependence relations When they exist among a set of functions Exercise Do this for l 7 t 1 t 2t using polynomial coef cients Do this for 1 cos2t sin2 using Wronskian style matrices or any other method you can think of A to V is linearly independent