N0910 - m 30H cum ~va cam/«=7 Ide 6‘ Fa c/“t‘o’tf...

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Unformatted text preview: m 30H cum ~va cam/«=7 Ide 6‘ Fa c/“t‘o’tf If MCx/dex + N(x,y) A) : o I} hit Q-Xac/E/ AM/VU/y] fut/k M )«Cx/y) MCx/yJOLx + MCf/yJNCXNJAyoa «3 Ovv\/€/)CG(JC' L’— dw‘. 9% 2’. 0m) 2 0% (WV) ay 9 Qfifl—rkaflgfl‘Nr‘tytfl! 3) a) 3X ax 9°7£N~MM= 911mm M 01X 5}— Jy 37 m g A Q 4, W )8 L/ 9.. X 4, E \c K? g 5) oLx /,< -* ,sz fiij: Q X :2 2X X (772+3’C) OLX + szjwoszo (9M. _’—— 3/” <9; .’—- ax 9) ,C;/CZ7‘1+3xljoLx+(7(jj : WHXSVCW N: : 1-er jcéjz ; WK)! : 2y1x4 3X7— 2.3 Solution Curves without a Solution Definition: Critical Point — A critical point (or equilibrium point or stationary point) of the differential equation dy/dx=f(x) is a point yo such that f(yo)=0 y=yo is a constant or equilibrium solution of the above differential equation. 5&2 heft poi/<2 k“ QXGWP’QQ Some conclusions about non-constant solutions y(x) of dy/dx=f(y) .A solution y(x) passing through the point (xo,y0) in region R remains in the subregion R for all x .f(y) cannot change signs in a subregion .A solution y(x) is strictly monotonic in subregion Hi .If y(x) is bounded above or below by a critical point c, y(x) must approach y(x)=c either as x+oo or as x+-oo If y(x) is bounded both above and below by critical points 01 and Ca, then y(x) approaches one of them as xeoo and the other as x+-oo S'éQUL‘v/K wa «MM fl EacflE: fl: P(a-bP) J 61,5 Mfg/«WW :fCP/ JCCPJLO =J Pia» Q/b 0" “Chad P04“? Wamwwgkg WW’Q’J’J’OF Pc‘M 3km}? cow, 6 WW Law S’ch 01“ 43V 9ij CHAPTER 3 : HIGHER — ORDER DIFFERENTIAL EQUATIONS 3.1 Preliminary Theory : Linear Equations 3.1.1 Initial Value and Boundary Value Problems Initial Value Problem: Solve: d”y dn—l an (x) dxn + an_l (x) y dxn—l d + .... ..+an(x);y+ao(x)y = g(x) Subject to : y(xo) = yo, y'(xo) = y1,...-,y‘"‘”(xo) = yH All initial conditions are specified at the same point x0 Boundary Value Problem (BVP): A BVP of second order Solve: d2 d W) de + a, (1031+ ao<x)y = M Subject to : y(a) = Yoda?) = 3’1 Dependent Variable and its derivatives are specified at different points May have several solutions, a unique solution or no solution 3.1.2 Homogeneous Equations n n—l an (x) Z; + a,,_1 (x) in} + .... .. + an(x)%1+ (to (x) y = g(x) If g(x)=0, homogeneous equation lf g(x)¢0, then n n—l an (x) if +an_1(x) in} + .... ..+an(x)i:l—xl+a0(x)y = 0 is called the associated homogeneous equation. .Differe’ntial Operators 9:5;/_D%, K 0” 017M , (V Q“CK))’ “l' - + alj,+Q0)/:o 'h ahD)+"" 4QrDj +ac\:_ ow (GAD-r +OuD—raojjzo LE ahDh‘l’ +Q,D+ao at szo .Superposition Principle for Homogeneous Equations lf y1,y2,....yk are k solutions of L(y)=0, then c1y1+02y2+....+ckyk is also a solution implications: - y=0 is always a solution - if y1 is a solution of the homogeneous equation, then c1y1 is also a solution 79W=o « ZX _ZX Sole—9.47013 yi’SQ )y12—6 /}3;o ’ )1: C,y,—iC2>/1+Q3 yLl-r 2(Qz—Y1L. QF-ZX) 5 avg: ' 0 4 watt, XJMM W; . 04‘ Nfiowxof yifiyz/S’oyb‘} FAQ 054199 Uge‘ Linear Dependence/Independence A set of functions f1(x), f2(x),.....,fn(x) are linearly dependent on an interval I if there exist constants C1,...,Cn, not all zero such that C1f1(X)+C2f2(X)+ ..... ..+cnfn(x)=0 " xii If the set is not linearly dependent, it is linearly independent. E71. /‘>SQ2X f 2. —Q:Zx/ jg : ZCQZK‘F 2"ij / L Cl7€+CL7fzfit+ 7’23} fa’k Clil, CZS—J“ / —2 C} 2‘ L Wronskian: The determinant fl f2 f" W(f1,f2,~--,f,.)= flu—1) 2(n—l) . H for-1) is called the Wronskian of the functions Theorem: A set of functions is linearly independent if and only if W(y1,y2,....,yn)10 for some range of x Definition: A set of n linearly independent solutions of a linear ODE L(y)=0 is a fundamental set of solutions. Theorem: The general solution of the homogeneous ODE L(y)=O is y=C1y1+02y2+ . . . . . .+Cnyn where yj’s comprise a fundamental set and cj’s are constants 3.1.3 Nonhomogeneous Equations lf yc=c1y1+cgy2+ .... ..+cnyn solves L(y)=0 and yp is a particular solution of L(y)=g(x), then yc+ yID is also a solution of L(y)=g(x) Theorem: If yc is a general solution of L(y)=0, the yc+ yp is a general solution of L(y)=g(x) 7c ‘} +jp‘kvq?’ (“251% “Le COMP’QQW‘Q’JC‘; IOQ‘Lttbn ya : (12 (2)2va yfieM-L‘wx of fatal/“6) y? .r 2 (PaAtKW-D‘M SoQuJIoh of 7/4 3)=/) :9 7:2+qu2x ? San—uliw 6F j/*9j’£6 Supp/was: k‘O‘M ye i} 5M‘O‘h 01L L9,}:0 3.3 Homogeneous Linear Equations with Constant Coefficients any(n)+an_1y(n-1)+. . ..+a1y’+aoy=0 i >1 4 0‘») : O y2C'Q’aX lvxokchJ:%> )wzmx 512 a Sdeuiidm 7(01 Rafi/kg 0% 5133 Characteristic Auxiliar E uation For the second order ODE ay"+by’+cy=0, the char. equation is: am2+bm+c=0 Let y : C'LMX >5] : C‘Mme y l/ : m a(C,m7QMX) +L(Ci‘"\€ X) + C(C‘IQMX to acid” (ahhr bM+C)20 W\X Somme amQ—FLM'FCZO C-E -_. GMQ+bm4Q2 0 .9 M: FbiiLQ'J‘rQL Zq Three Cases: 52—11-QC.>C> PMIQMQ’GAQMWOLM; f “C 9&3: yuLtacus M, 5M1 aMmO-‘QWQM Coin; 51—I+CL¢<0 WM 9 ML are CW/QQX ...
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This note was uploaded on 07/25/2008 for the course ME 391 taught by Professor Blazer-adams during the Fall '08 term at Michigan State University.

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N0910 - m 30H cum ~va cam/«=7 Ide 6‘ Fa c/“t‘o’tf...

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