N0827 - 98/27 CLAS‘J‘ we're; mi; 361: 1.1 Definition...

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Unformatted text preview: 98/27 CLAS‘J‘ we're; mi; 361: 1.1 Definition and Terminology Differential Eguation: An equation containing derivatives of the unknown (dependent) variable(s) with respect to one or more independent variables. Non—oincfemfic—Q Ewb‘oh fo’x 3CX).’}IZ+>’X ~56nxzo ‘DULJLWRJ EM‘w: 0‘9 Z A + cos X :2 dx y 0 Classification of Differential Eguations Differential Equations can be classified by: 1. Type 2. Order 3. Linearity 1. Type: Ordinam: An ordinary differential equation (O.D.E.) has only one independent variable, e.g. time t or space (1-dimensional) x However, it can have more than one dependent variable. Partial: A partial differential equation has more than one independent variable. \cvxckpgmM 6 _._.+§I.: C9 0, Mum Mai/0y Lyl/yz gyf'LQ/M 01L Okk ‘ ocvdaHm—y fly‘ +y'ytzo l {\JQPDJAM VGTZL‘Qb/QQ.‘ 1; EXaMlfi-A’ or 9):} , 9"” +91 :0 l Miami. r7. 3 7, r dx >’ Z \AWMt Z ’( L T 83x1 Jeaglid 35.1%,; iii—Lo \ Mum T Li \WMM k»; is L Aside f NUE action LECLKCZ NEQ'EL‘OVK \ Qty A1 J J \ Md “\WMM V‘ahk‘al’lfij’ 2‘ ?9'7Cw \a‘ka'kfofl 9/ 2. Order: The order of a differential equation is the order of the highest derivative in the equation. Therefore, we can have ODEs of first order, second order, third order, etc. ll/ y «LC/93+?) 95‘me ‘V 3’“ 01:14am Hf 0% {—W Hcgkujc dwvc/fnre Cry/7 :9 OHM 0F H‘Q 63E u 3 N°£ HQ 0AM (VF I L OnAMQF‘rke ODEQ 3 f "l v\ yl+wy;o "a ZAolLN ODE 2/ it“ "7‘ 'jfoté‘koDE t General form for an n order ODE in one dependent variable: F(X!y!ylgy”, . . . . ..,y(n))=0 n where F is a real-valued function of n+2 variables: X,y,y',y",y"',....y( ) n Normal Form: The above equation can be solved for the highest derivative y( ) in terms of the remaining n+1 variables, giving us the normal form NO’LYvLa/Q *ohm \, 05:1 fl ow P f house-MW) Classification of Differential Eguations 3. Linearity : Linear ODE: An nth order ODE is defined to be linear when n n—1 an(x) y( ) + an-1(X) y( ) + ....... ..+ a1(x) y' + ao(x) y = g(x) n — The dependent variable y and its derivatives y', y", ....y( ) must be of the first degree n — The coefficients of y, y',....,y( ) i.e. a0, a1, .... ..,an are either constant or depend on the independent variable x. Imp; A linear equation cannot have non linear functions of the dependent variable or its derivatives; e.g. cos y, in y, etc. make an ODE non-linear Non Linear ODE: An ODE which does not satisfy the requirements of a linear ODE is non— Hnean Exampfl‘bf of Linear. Evvatiaty \__ XN (D dfi + 3x1 at): X' Av" '01? Hoflrg ~3Mk 031019”- kaflah ODE @) (y—xmzx 4Ll-X at?) :0 25Lij er Jul/oab‘om to (F3- hohwsnfl 7La>wn dz 4 e biz : y Y) ’ "i +35: 0” LrX \+X My («fly C3) yI—Zy—iQXSO _ \St OJ‘M ODE EXOUKPJQ O’F' Non LEW CD'S} 0%) ’+ 2y >3 Wy (I Ck AAAC‘L ton ofy/so hoanemzom ODE {:LQ,“ q 91% CUMLL‘VVJLEUK 01L 9: W70. me Q.y,Cx)+qsz(xj Mufi: C‘QIQ Sovqke HQ QCVUcd-(oh m F (x, a.y,+quu sly/+41%) - , qmchl+qzyz<ww LL Hm ahuV-e W'+fo-\ 0 “0% 5636573104 J HQ ODE\} ham—Q‘kecoz Classification of Differential Eguations 4. Autonomous vs. Non-Autonomous Autonomous: Independent variable does not appear explicitly in the equation I II F(y,y,y , .... ..,y )=0 Non-Autonomous: Independent variable appears explicitly in the equation *2ng we y/Il+ gyfl+yfz 2 / yI/+CJQ :0 EKOWQ/QM’ ml m —du{bwwuj DJ? 11/ 43y’2-t5j E2.) CL;— 0% -vaE I Special types of linear ODE's an(x) y(n) + an—1(x) y(n_1) + ....... ..+ a1(x) y' + ao(x) y = g(x) Definitions: Constant Coefficient Linear ODE: If all aj(x)'s are constant Homogeneous Linear ODE: If g(x)=0. Such an ODE has y=0 as one solution EXQKP/QH 01L COQ7L2L5C‘CAM‘L‘ Lt'kaan ODE 3fz+j2 3x ) ym+ Lrj7+y;l EX amp 119:? O 12 HOW Zzheouj Li Him ODE ijI//+ jfl+ 3y=0 Solution of an ODE Solution of an ODE F(x,y,y',y", .... ..,y(n))=0 Basically, ¢(x) is a solution if F(x,<D(x),¢'(x),<l>"(x), .... ..,<D(n)(x))=0 ¢(x) is defined on an interval | of the independent variable. This interval is called the interval of definition, the interval of existence, the interval of validity, or the domain of the solution. MoSIC QOWQLVOH COLKME b—Q SOQVQOk Ch QROSJLOL fakafiyptraa M “ULNJM‘Co/Q Screw—hm ii +aj:o W azcotxy’mflvf: x @M SOMM O )20 dcoj + 0(0) :0 (ix Awailux mm‘m t5 )1: 241K y/:'_ag,ax fiat; any ~ zae’axfit (kg—ax My ...
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N0827 - 98/27 CLAS‘J‘ we're; mi; 361: 1.1 Definition...

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