Class Note for MATH 254 at UA 2
Class Note for MATH 254 at UA 2
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Date Created: 02/06/15
MATH 254 Course Summary Trial Final Trial Exam Course Summary 1 De nitions7 Classi cations Section 11127 Lectures 12 Type ode7 pde7 system of ode7s7 pde s For ode7s order7 linear or nonlinear7 autonomous or nonautonomous Solution lmplicit Explicit How to nd latter from former lnitial Value problem First order ode s 1 3 fxy 7 CI y 7 numerator 7 Fzy 7 denominator Methods Direction elds SLOPES l37 Lectures 127 34 Linear 7pzy lnt Factor 237 Lectures 56 Separable fzy 22 Lectures 127 34 Autonomous Graphical Lectures 34 Notion of equilibrium point ye 0 Lectures 34 Lyapunov stable7 unstable7 asymptotically stable equilibrium points Basin of attraction Stable amp unstable manifolds Lectures 34 dy g fzy7 yzo yo l2 Lectures 12 Applications to population growth7 radioactive decay7 salt mixing7 heatingcooling7 falling bodies7 LR amp RC circuits Chapter 3 Numerical Euler scheme l4 Lectures 567 710 Existence amp Uniqueness of solutions to 2 Linear 2nd order ode s lt2gt ltzgt ltgtiltgt 61 I dIQ p dz 419 9171 71774790113011 See Chapter 4l7 42 437 44 457 46 Notes on complex arithmetic7 487 49 Lec ture Notes 11157 1621 General solution of 2 general solution of 3 y pzy qzy 0 particular solution of General solution of 3 lin combination of 2 li solutions of Notions of 1d and U of functions on intervals and the connection With Wronskians Equivalence of Li 1d of y1 yn solutions of 4 and a1y aoy 0 and nonvan ishing vanishing of Wy17 yn the Wronskian of y1 yn For 3 Wy1y2 Wy1y2 10 exp 7 f psd3 Methods for solving 374 With constant coefficients and l With constant coefficients and periodic forcing Undetermined coef cients MassSpringLRC circuits Notion of resonance and tuned oscillatorcircuit Applications Variation of parameter and reduction of order methods 487 497 56 3 2nd order autonomous systems amp phase plane analysis Chapter 5l567 Lectures 2230 Equilibrium solutions rage of 5 dz dy 7 a Fzy E Ciyy Their stability Lyapunov7 asymptotic properties7 basins of attraction7 stable and unstable mani folds d C Connection With solutions of 67 i 17 M F I y Applications to pendulum7 predatorprey Use of SLUPES EL SYSTEMS Qli Type ode7 system of ode s7 pde7 system of pdels the following differential equations For ode s7 give the order of the ode or the order of the systerni dQI 1 W 352 0 dQI dz 2 W2E5010 dzi dyi lt3 37y afayism What is the connection between 3 amp 2 lt4 amz dy d2 7T17y7127E77b2zy7 a 7 b are positive constants Lorenz equations ltsgt m lt6gt ultzvtgt2ezltzvtgt 2 7 zt 131 uz t 7 uztu2zt zt 132m uz tvx t 8 Given uzyzt7 vzy2 t7 wz7 y 2777 pzy2 t7 p is constant7 1 is constant Bu 31 311 EJVEJVW O uvw vltg 2 uv wg 7 vlt gt uzy2t By showing 1 exist for all time or by showing they don7t7 you could win you 1M H w See htthwwwiclayrnathiorgMilleniurn PrizeProblernsi 3 Q2 For each ordinary differential equation below7 give its order and state Whether it is linear or nonlinear LNL7 autonomous or nonautonomous ANA Equation Order LNL ANA lt1 2 ii 27zitcosz0 iii 2351 0 iv 47 i 0 v g t2 1 vi 21 27 i 0 vii 1 7 icoswt dt2 m dt viii 327 y 0 327 1 0 x 2 z xi d2i0i2z271ddiz Solve i With iici 11 2 1V given 10 7 5 0 71 71700 7 4 3710 7 7 v With iici 10 2 vii Solve for 7 2 g 7 5 g 71 10 7 0 7f0 7 0 viii given y0 71 0 7 2 ix given y0 l 0 0 x given rm 7 ltogt 7 71m 7 57 0 7 0 Xi by Writing it as a system and using SYSTEMS Extra dif cult iii Solve given 10 E 1 120 V 0 0 Where V is arbitrary vi Solve given 10 E d V 7 E Q31 Solve the following initial value problems lt1 21y 7 ya 7 2 ii M17 y y0 2 iii dy 7 7 E y3 74 Find the branch through 3 74 E y iv 27 y0 721 Find the branch through 0 72 V yZO Find all solutions With y0 0 Find all solutions With y0 11 vi ylny y0 6 vii 377250170 10 71 7f0 70 viii T 01291011cos3t 1000 ix 717161 10 070 x t2 Tgt 7f7z7t 11 70 7133170 xi 12 1 7 21 21 7 2 10 7 7f0 7 01 xii 121ng 7117 2 317 7 t 21 7 213 11 7 0 d3 7 01 Q4 Using graphical methods discuss the long time behavior of solutions to 7 7117 z1 z2 7 Identify a the equilibrium points b say whether each is stable or unstable c say if stable whether asymptotically stable d identify the basins of attraction for the asymptotically stable equilibria Q5 At time t 0 a 200 gallon tank initially containing 50 gallons of pure water is lled with a brine solution containing lb of salt per gallon at a rate of 4 galsmin It is drained at the rate of 3 galsmin and the tank is kept well stirred throughout 1f zt lbs is the amount of salt in the tank at time t minutes write an ode for What is 10 What is Ct the concentration of brine in the tank at time t in terms of zt and the volume Vt of uid in the tank at time t What is Vt After how many minutes will the tank over ow Show that C0 0 C50 73 C100 g and C150 lbsgal Draw the graph of Ct vs t for 0 lt t S t1 t1 over ow time Q6 A cake is taken out of an oven at 300 F and allowed to cool in a room with temperature 60 F At 3pm its temperature had dropped to 160 F and an hour later had dropped to 110 F At what time was it taken out of the oven Q7 Consider the two tank arrangement shown in the Figure below At time t 0 the valves A B C D are simultaneously opened Find the amounts of salt zj t j 1 2 and concentrations Cj t j 12 of salt in tanks 1 and 2 as a function of time Draw the graphs of C1t and C2t A 2 gals min 7 1 lbgal Ill initial volume 200 gals initial concentration ilb gal initial volume 100 gals initial concentration lb gal 7gt 0 Tank 1 B Tank 2 lm 2 gal min 3 gal min Q8 A body of mass 10kgs falls from rest under the joint in uences of gravity 9 10 msec27 approxi mately and friction the friction force is 001 times the square of the velocity Show that the velocity after 51n2 seconds is 73333 msec We measure zt and vt if 7distance and velocitypositive upwards To what value does vt tend as t 7gt 00 Calculate z101n 2 Q9 A very light spring its extension under its own weight is negligible hangs from a hinge A weight of 2 kgs is added to the pan and the spring is stretched by 2 cms If g gravity is taken to be 10 msec2 and if the friction force experienced by the massstring arrangement is numerically equal to 4 times the velocity nd the equation which the displacement from equilibrium zt satis es Given 10 1 cm and 77 0 msec write zt in the form Ce quot cosbt 7 go Determine C a b and cos g0 sin g0 Q10 Consider the LRC circuit shown below S amps A B L C qt coulombs R Given V 7 VA 7V0 coswt E D and q0 0 0 At time t 0 the switch is closed and the emf starts current owing in the circuit Solve for both qt and and in particular show that 6 quot A1 cos bt A2 sin bt V i0 coswt 7 so Write expressions for ab and cos g0 singo in terms of LRCwi Also nd A1A2i For L C l draw the graphs carefully of m versus w for the values R l R 2 R 1 At What value of w is maximum What is its value there For this value What is VoZ coswt 7 go equal to Q11 Consider the 2nd order autonomous system dzi dt y dyi 3 E77zzi What are the equilibrium points 18 ye Are they stable or unstable lf stable are they Lyapunov stable or asymptotically stable lf unstable nd the equation for the graphs of their stable and unstable manifolds near rays The solution curves ztyt in the zy plane obey the rst order ode dy77zzg dzi y a By direct integration compute the integral curves Write expressions for the stable and unstable manifolds through the unstable equilibrium points 18 ye b By hand sketch out direction elds namely calculate 9tant9 7 for several points such as 07i7 07i2 i707 igyob ilyil c Use SLUPES to do the same thing Use SYSTEMS to nd solutions ztyt starting out With initial conditions 1 10 0y0 i5 ii 10 0 y0 i1 iii 10 0 y0 if Q12 By adding a damping term to Qill7 d1 dt y dyi 3 Eiibyinrz With b 1017 recalculate phase plane Q13 Draw the phase plane for the undamped pendulum d2z w slnz 0i KNOW THIS WELL You could explain this to your mother using a meter stick7 right 39 d 7 H1nti Set Ta 7 yr Q14 Consider an an71 gTii a1 f7f aoz 0 Given the following roots of the characteristic equation an39r a1quot a0 07 Write down the general solution of a T00011717i71i b T11112712i7172i712i7172i c 7 07027223977239723977239 d T12345i7i Q15 Find the general solution for d2 d a g tg 0 b 721 mgr cos wt if Lug c ng DJQI cos wt 3 2 d 73 371e 4 lta4 Hint the roots of T4 71 are 7 Hi Check 9z g Given 10 0 0 0 1 in Show that zt iew 7 ie w 7