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# 218 Review Sheet for MA 26600 at Purdue

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 31 views.

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Date Created: 02/06/15

MA 266 Fall 2000 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 fpolymonial dz f e du f u du including 1 1 am b dz partial fractions X Xx T2 You should be able to use given values yzo ya 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 field HOMOGENEOUS EQUATIONS dy y 2quot 2 d a y 1 3 IetyU mv odx a vandx 2 Substitute to obtain v Fv Solve the above separa 1 equation for v in terms of X Substitute v g to obtain a formula for the solution y of the original homogeneous equation EXACT EQUATIONS MOE Maw o is exact if Myc y N1 y Find a functional 2 y such that 21 2 y r M 9 Y and 11x y N z y Wm y f M x 3 dz hzzs01ve 5y May dx h y N my for hy A solution y yxof the exact equation then satisfies d d d y 11223 11 10m 21 Ma y Nx 11qu 0 so the general solution is of the form 1M3 yc Numerical Methods For Solviny f t y yto yo Create an M le to define the function ft y The function name and the le name should be the same Note that Mfamps are not entered in the matlab commandwindow but are external text files that are created with a text editor EXAMPLE If y 2 55 create an Mfile named f11m function zf11ty zsqrtt39 The general syntax for the Euler tangent line method is gtgt 13 eul d le t0t naly0stepsize Note that stepsize t nal t0 n Where n is the number of steps EXAMPLE To find the Euler tangent line approximation of the solution of the initial value problem y t 111 8 where t 2 using stepsize h 05 quot gtgt tyeul f11 12305 gtgt ty 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 runge kutta method use rk4 in place of eul To obtain the graph of an approximate solution on a direction field enter gtgtplott yC where C o x Omit C for a connected graph APPROXIMATE SOLUTIONS The matlab commands eul rk2 and rk4 can be used to obtain approximate solutions of theinitial value problem 31 fxy yao yo You should be able to use the formula ya Un l fxn 1 Ila 0h to evaluate Euler tangent line values by hand Approximation methods may not give good approximations of the solution of the initial value problem 3 fay 3130 yo if c The initial value problem does not have a unique solution because either f or f is not continuous at the initial point 0 The approximation extends beyond the interval where the solution is valid because either y t or ya becomes unbounded o The solution is unstable because solutions that have slightly different initial values divergafrom the desired solution PROPampTIES OF SOLUTIONS If f has continuous first partial derivatives then solutions of the differential equation 1 f x y satisfy fatty fyxw f2zry fvyfy If 93 is a solution of the differentiable equation 3 f x 1 If y gt 0 at a point then y is increasing near the point If y lt 0 at a point then y is decreasion near the point If yquot gt 0 at a point then y is concave upward and the Euler tangent line approximations are less than or equal to the solution near the point If y lt30 at a point then 3 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 aboutxCis fltgtltgt fcf39cltx ci z c2591ltxc3 gs cw MA 266 Fall 2000 REVIEW 3 PRACTICE QUESTIONS 1 Determine whether each of the following differential equations is separable homogeneous linear or exact Brie y justify your answers d a Zxyz3yamp0 bx3y2my gzo c x3y12 y1 0 2 d d 2xy1 a 1amp0 dy 2 211 1 yz1I a 2 Find the explicit solution of the initial value problem 3 y2 1 y0 0 3 Findthe general solution of the differential equation xy 2y 22 d 4 Use the formula y am to express the differential equation dz in 2 terms of 31 and 5 Find an implicit form of the general solution of the differential equation d3la2y2 dx my 6 Find an implicit solution of the initial value problem d 2zy 1 x2 23035 o y1 1 7 Determine approximate values at x 05 of the solution of the initial value problem y 3x y y0 1 by using the Euler tangent line method with I 025 8 Use the given direction fields and the graph of an Euler tangent line ap proximation of a solution of an initial value problem to explain why the ap proximation is not a good approximation of the solution 2 a y39 32123 y0 o b y y1 o um I 39lIrlfl I I I II I I34IElI39l I I 4 IIIyIIIII31IIIIIIII i A 9 i r 39i1 wt IIII39IIIIIIIIIIIIII I M 11 139 Iquot r 15quot quotIquotIf 1quot7339139 IE I39 r I 39r quotIIIIIIIIIIIIIglriI 39II 7lllglllllllll It gtIIIIII5IIIIIEIIIIiI 39IIIIIIIIIIIIIIII If 11III39III1IIAIII IIIlIIIIIIII 17quotquot u 1 4 1311 I Iquot I I 1 It IIIVIIIIIII II111ll ullt443t At4 t 2ll I z 1 III vyrzrglrf lilI u lZ l lllil llIl I III IgI z Allirill I Jul Al 114 er aafaa aa I I 1 l l u 0 00 Qquot 0 07 u I i c y y2085y0 1 d y 103 llequot y0 1 ym runum 7 n Iillll v z4i IljagIoslIJI III II II J X 7quotrr IIIlJIJIJI Dill lglgt a IiItlt uquot39 x0 39quottt s u II I I gl I l ll l L 111 I I t t quot quotquotquot V j 39 3939E quotSquot39quotVquotVquotlt3939quotampquot IIIIIIIIII IIIIlleII E39oxxs I I 39I I gr I I I I I I l I I I I I i I quot1quotVquot Jquot quot39lquot39Cquott quot39quot quot K39T YKquot ltquotlt lt quot1 I I I I I I I In I H II I I I I 1s MS3 3 a i IIIIEIIEII II I ll Elbli E39Sampquot g quot quot 39Vquot IIIIyIIIIII III r511 u 5 MIVIIIV lIIIIII NIHIW 1111 1Ats I I f I I I I I I I I 7 1 I39 I 11 j1amp431A1 L Ill111 I 4 1 quot llIquot I I ll u il ili4 a 4Lsgu agtEi 9 1 t5i TT 1 H hIq Imrw HI 4 I I I rz omtuuuuuuuuu 390 Hanna an 9 Consider the initial value problem y my 32 y8 1 a Is the solution increasing or decreasing near a 3 b Is the solution concave upward or downward near X 3 c Are the Euler tangent line approximations of the solution near x 3 greater than or less than the solution 10 Find the rst four nonzero terms of the Taylor series about 6 1 of the solution of the initial value problem 3 223 y1 2 MA 266 Fall 2000 REVIEW 3 PRACTICE QUESTION ANSWERS 1 a homogeneous exact b homogeneous c none of these types 01 linear exact e separable exact 1e2 2 y 162 22 C 3 y7 4 112131 1v l v dz 1 y 2 1n C 5 2 In 6 zzyxy2 1 7 y 175 8 a The initial value problem does not have a unique solution near 30 The functions yt O and yt t3 are both solutions b The approximation extends beyond where the solution is valid The solution approaches a point where y x 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 9 a 51yN 3 1 implies y my yzw 2 lt 0 so y is decreasing b 33 It 3 1 implies y xy y 2311 m 3 lt 0 so y is concave downward c The Euler approximations of the solution near x 3 are greater than the solution 10y22x 1 3x12 8x 13

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