!"#$% &'()*!*$!% k + pk + q = 0 $ !+, #,'*-%$ 2 k1,2 p −p ± p2 − 4q . = 2 ✭✶✳✷✮ 3,4"5()6-- #,'*$ k1,2 ✱ *(84!% ,9:!! '!;!*$! ,4*,',4*,+, &'()*!*$- #(# "&✲ =!'=,>$?$@ 4)&A ("5*BA '!;!*$8✳ C),8"5), "&=!'=,>$?$$✱ #,*! *,✱ ()5,%(✲ 5$ !"#$ +('(*5$',)(*, 5!%✱ 5, $"A,4*,! &'()*!*$! -)6-!5"- 6$*!8*B% y(x) = C1 exp(k1 x) + C2 exp(k2 x), ✭✶✳✸✮ +4! C1 ✱ C2 − =,"5,-**B! )!6$ $*B✱ ,='!4!6-!%B! *( (6E*B%$ &"6,)$-%$ y(x = 0) = y0 , ′ y (x = 0) = ′ y0 . ✭✶✳✹✮ ➘6- >(4( $> #&'"( %!A(*$#$ &"6,)$- ✭✶✳✹✮ =,4'(>&%!)(@5 ,='!4!6!*$! *( (6E✲ *,+, =,6,H!*$- $ *( (6E*,8 "#,',"5$✱ ( ) >(4( (A '(4$,5!A*$#$ − )!6$ $*& I6!#5'$ !"#,+, >('-4( $ 5,#( ",,5)!5"5)!**,✳ C 5, #$ >'!*$- %(5!%(5$#$ "&:!✲ "5)&!5 5'$ '(>6$ *BA '!;!*$- $"A,4*,+, &'()*!*$- ✭✶✳✶✮✱ ,56$ (@:$A"- %!H4& ",9,8 ) ",,5)!5"5)$$ " 5!%✱ #(#,8 )$4 $%!@5 #,'*$ A('(#5!'$"5$ !"#,+, &'()✲ *!*$- ✭✶✳✷✮✳ J(""%,5'$% I5$ "6& ($ =, ,54!6E*,"5$✳ C6& (8 ■✳ ✃,'*$ k1,2 4!8"5)$5!6E*B!✱ *! ",)=(4(@5 4'&+ " 4'&+,%✳ ✃,'*$ k1,2 -)6-@5"- 4!8"5)$5!6E*B%$ $ *! ",)=(4(@5 4'&+ " 4'&+,% ) 5,% "6& (!✱ #,+4( 4$"#'$%$*(*5 A('(#5!'$"5$ !"#,+, &'()*!*$- 9,6E;! *&6-✳ ➶ I5,% "6& (! p − 4q > 0✱ ( ,9:!! '!;!*$! &'()*!*$- ✭✶✳✶✮ $%!!5 )$4 2 # !p #$ ! p p p2 − 4q p2 − 4q y(x) = exp − x C1 exp − x + C2 exp x . 2 2 2 " ✾ ✭✶✳✺✮