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# 242 Review Sheet for MA 26600 at Purdue

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Date Created: 02/06/15
Math 266 Spring 2001 7 REVIEW 3 FIRST ORDER DIFFERENTIAL EQUATIONS You should be able to recognize and know how to solve rst order differential equations that are either separable linear exact or homogeneous You should be able to evaluate integrals of the following types polymom39al das f e du fu du including 7quot 71 f 2201172 das partial fractions You should be able to use given values ya0 go to determine unknown constants in a solution You should know the relation of the graph of the solution of an initial value problem to the corresponding direction eld HOMOGENEOUS EQUATIONS Ly F 3 dac ac Letymsoaijvandu Substitute to obtain 06 v Solve the above separable equation for v in terms of ac Substitute v g to obtain a formula for the solution y of the original hornogeneous equation EXACT EQUATIONS May Ncy 0 is exact if WE93y Nxcy Find a function 11193 y such that 111495 y May and wy y Ncywa y f Ma y 6156 My solve 8 May dx h y Ncy for hyA solution y of the exact equation then satis eswxy 111495 wyxy May Ncy 0so the general solution is of the form 11cy 0 Numerical Methods For Solving y fty yt0 yo Create an Mi le to de ne the function fty The function name and the le name should be the same Note that Mi les are not entered in the rnatlab command window but are external text les that are created with a text editor EXAMPLE If y t y create an Mi le named f11m function Zf11ty Zsqrtty The general syntax for the Euler tangent line method is gtgt t7yeul7d le t0t naly0stepsize Note that stepsize t nal 7 t0n7 where n is the number of steps EXAMPLE To nd the Euler tangent line approximation of the solution of the initial value problem 7 415 7 y1 37 where t 2 using stepsize h 05 gtgt t7yeul f11 1727305 gtgt lt7yl ans 10000 30000 15000 40000 20000 51726 The syntax is the same for the improved Euler method use rk2 in place of eul and the rungei kutta method use rk4 in place of eul To obtain the graph of an approximate solution on a direction eld7 enter gtgtplottyC where 070777X77Ur7 Omit C for a connected graph APPROXIMATE SOLUTIONS The matlab commands eul7 rllt27 and rk4 can be used to obtain approximate solutions of the initial value problem 7 ay ya0 yo You should be able to use the formula yn ywl facn17yn1h to evaluate Euler tangent line values by hand Approximation methods may not give good approximations of the solution of the initial value problem 7 ay ya0 yo7 if c The initial value problem does not have a unique solution7 because either f or fy is not continuous at the initial point 9 The approximation extends beyond the interval where the solution is valid7 because either 705 or becomes unbounded o The solution is unstable7 because solutions that have slightly different initial values diverge from the desired solution PROPERTIES OF SOLUTIONS If f has continuous rst partial derivatives7 then solutions ofthe differential equation 7 ashy satisfy 9 MW fy0c7y MW fyI7yfI7y If is a solution of the differentiable equation 7 ay If y gt 0 at a point7 then y is increasing near the point If y lt 0 at a point7 then y is decreasion near the point If y gt 0 at a point7 then y is concave upward and the Euler tangent line approximations are less than or equal to the solution near the point If y lt 0 at a point7 then y is concave downward and the Euler tangent line approximations are greater than or equal to the solution near the point The Taylor expansion of f about cc c is x fc f ca 7 c 7 c2 7 c3 MA 266 Spring 01 REVIEW 3 PRACTICE QUESTIONS 1 Determine whether each ofthe following differential equations is separable7 homogeneous7 linear7 or exact Brie y justify your answers 7 395 8 a 20yc3yj0 b x3y2xyo cc3y1295y10 d 2xy16210 e x21y210 Find the explicit solution of the initial value problem 7 y2 7 17 y0 0 Find the general solution of the differential equation 567 2y 02 Use the formula y m to express the differential equation 3 g in terms of 557227 and Find an implicit form of the general solution of the differential equation 1 Find an implicit solution of the initial value problem 71 Qxy 1x2 29 07w Determine approximate values at cc 05 of the solution of the initial value problem 7 y7 y0 1 by using the Euler tangent line method with h 025 Use the given direction elds and the graph of an Euler tangent line approximation of a solution of an initial value problem to explain why the approximation is not a good approximation of the solution 2 23 7 7 3t 7 809737 774070 byiayH 1 0 V Wm v ewe 6l4 l4 4l wImhll lfl Orr wr w zy zr iiiyiyiyyyiiylI y A 31ryrvrlirrr1 35 g iiiHHHHHHH NM NM 7Illlllllllllllll 1 1 2fll Ifl lf39TYVIT lVfl zlflV I T Illllllllll lllll llllll i1vJlv1 illllldl lvlll HHHHHHHH will HUN gt UHHHHHUH m gt VHHVH HM nllllllll u H in ll l HHHNHHHHHH Him HHH r 77 llllv il il ll lll Vgtr llllll llllll 2 KK rzllIK IIll NHNNHHHHNH NH 1 HHHHHHHH HUN HUN 393 3937 NHHHHHNNH NH HM 7HMNMMHMNM l VHMN Mr h I 10y 711847 y0 1 d 1 920857 90 1 C y xxxxxxx 4 iiiiiilt wwwwwww yiiiiii yyyyyyy yii yyyyyyy riivv xxxxxxx lt iiiiilt xxxxxxx iiiiiiiii xxxxx viixxxiiiiirii yyyy yrxxxxxiixiii 11111 my iii gtgtgtgt rxxixviriiiirii iiixx xxxxxxxxxxxx x 14 wwwwwwwwwwwwww ii n ltltltlt zl4lllilt ill5147 xxxx l ll zli 44541Jl4 yyyy ill44447711 l 4 4 Tllil ErLriiiw 47 l gtgtgtgt xvirirxlll yyyy 1714 47111111 Ill4 47le rr xjrr L 4 4 Tlll ivrlrz4r4riIrv ylrlr144x 1111 14 lily14455le yyyy 44yy r 4 4 72le lti1wxI rilt lll 4 4 77114 1111 iz iir lvi gt 9 Consider the initial value problem 7 any 7f y3 71 a Is the solution increasing or decreasing near cc 3 3 b Is the solution concave upward or downward near cc c Are the Euler tangent line approximations of the solution near cc 3 greater than or less than the solution 10 Find the rst four nonzero terms of the Taylor series about 0 1 of the solution of the initial value problem 7 5527 y1 2 Math 266 Spring 2001 REVIEW 3 PRACTICE QUESTION ANSWERS 1 a hornogeneous7 exact b hornogeneous c none of these typesd linear7 exact e separable7 exact 1621 16 27 4394 L2 4 37 49 44 10 dacvi 171 495 6 x2yxy21 7 yg 8 a The initial value problem does not have a unique solution near 00 The functions yt 0 and yt t3 are both solutions b The approximation extends beyond where the solution is valid The solution approaches a point where y a becomes unbounded c The approximation extends beyond where the solution is valid The solution approaches a vertical asymptote d The solution is unstable Solutions that have slightly different initial values diverge from the desired solution Pictures v name Wm V m xizilliiviiiizzxxr Cxr iiiiiii i szvrmiiiiwv1xx iiiiiiii r mmwm V n n I1zzzzxr r 111ri rrrrrr xx rrrrrr rr 7 1 zyr wimrrVJ C rrrrr i V7Inss illz izzlll hskipltruein d 56y y2 m 2 lt 07 so y is decreasing 9 a my 3771 implies y b 5077 x 37 71 implies y 507 7 ny 73 lt 07 so y is concave downward c The Euler approximations of the solution near cc 3 are greater than the solution 10y22x71395712395713Hgt

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