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Math 431 - Math for Science and Engineering 1 Section 2.1 Homogeneous Linear ODEs of Second Order “4 Seconid Order Linear ODE: second order near ODE has the frm af ald + oe) = 702) ts Nenlarr if eaanot be ten a this orm, Te Homogensos 765)°9 xampie 4, Siate some examples of xeon order ordinary diferent equations i ; Ags 37 gy FOS = 2" ines Ins Oar Trai sc Seen’ 3)" sag linent Spee s tex oraee My FF Non hwwtens J =O Wanlihesy LN. OVW SB =o wnenr gyser 2 Sa Frienss vice = ——— > [et achat an ODE na to = pean tah) \ tna and Homegr__| Tagen x €, noe thir ay works ‘ak uti se on Gee Gaon QU 7c amt he hat Lins Poi does sanees! pyr al Fe gost ERASING FLX 4 = De sinwis\Y Ay Peale 5 ges be he OD y= Danae nk eel Prieiple dors hold dt 7 HIG 3S BF OB Yay) = 19a toy Meee av £5/4 “jnittat Value Problem, Basix, General Solution, and Particular Solution yt seed aleyy=2 kee) an VG) =e or loo) =n ula) Has a general solution Math 431 - Math for Science and Engineering T tant Conficients Section 2.2 Homogeneous Linear ODEs with Const Homogeneous Linear ODEs with Constant Cooficents Toth act ill consider yeoyew=0 to te Homonaneows near {Ty the ston to ind the pera rom of shins (ODES with constant coeicents +6e™ li awe “+dKby= 0B Basis for General Solutions: “Fhe basis forthe general solutions aie as follows base off the roots of the characteristic nquation: et oe nn ea iin i ene yeoo tot Faure vivewomne ar Sileel Clo. et: PEAY ee Ds SreP DX deny cin Dee pe, Gea ye ONG. “i > -S ite, Adore 3] €, 25 if poston te —_____ nt & Corer comple ols X= 0 roains 5) 2 Dead Sambo 10 di” Fen) [able S OSH sting tM Re sO} FiGyy to the fallow teat \ fle converting complex tution n= e7sin(B2) = sin( fe) ee puis poo © with y(o) <0 ana ¥(0) —9 Seep \ Vs bu a ey ae ere ae Aone SPD gee pony @ Seay OSB)! ce? Sing Sin =h= © ot PASI SIAHERIE (Cie 9) ey €,=0 op B'2 BG egs@y - 6262 sin) 7 ~bz, 0) B= age eaq—orce ~~ 
