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uhsluhsl

@NH@JY[NP MGPA@NP KJ BGP J@ZG@ANHJPKADJ]H@AGBJP N]KAHG]AGP

BA@. [JBLN SGTGBJ[G ]NPGP

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Kjdha`aðh (7)Kjdha`aðh (7)Zhg j`ug`aðh kajrjh`agb (JK) js uhg j`ug`aðh qujahvnu`!g "j!avg"gs "j uhg #uh`aðh "js`nhn`a"g "juhg n vg!ags vg!ag$js.

JcjlpbnJcjlpbnBgs sa%uajh&js j'(!jsanhjs snh j)j*(ns "j JK+s,

9=0=

=

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@BGPADA@G@AÐH3 J@ZG@ANHJP KADJ]JH@AGBJP @BGPADA@G@AÐH3 J@ZG@ANHJP KADJ]JH@AGBJP 

  Bg j`ug`aðh kadjrjh`agb `nhtajhj kjravgkgs

Nrkahgrags kj uhg n lçs vgragmbjs kjpjhkajhtjs `nh

  rjspj`tn g uhg snbg vgragmbj ahkjpjhkajhtj.

[apn

Bg j`ug`aðh kadjrjh`agb `nhtajhj kjravgkgs

Ygr`agbjs pgr`agbjs kj uhg n lçs vgragmbjs kjpjhkajhtjs.

 

Yraljr nrkjh  D( x, y, y»)< 9

Pjfuhkn nrkjh  D ( x, y , y», y»»)

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  g) Bg vgragmbj kjpjhkajhtj  y y tnkgs

sus kjravgkgs snh kj 7jr. frgkn.

  Bahjgbjs   m) gkg `njda`ajhtj kj y y sus kjravgkgs

  kjpjhkj kj snbgljhtj kj bg vgragmbj ahkjpjhkajhtj x !pujkj sjr `nhstghtj).

 "rgkn

 

 #n bahjgbjs Bgs $uj hn `ulpbjh bgs prnpajkgkjs

ghtjranrjs.

 

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%&

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 y

u

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Jh bg lgynrìg kj bns bamrns bgs kjravgkgs nrkahgrags sj js`ramjh `nh bg

hntg`aðh kj Bjamhaz, n majh, `nh bg hntg`aðh kj Yralg

 xj ykxky

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J@ZG@ANH ADJ!JH@AGB N!AHG!AG

Pa bg duh`aðh kjs`nhn`akg kjpjhkj kj snbn uhg vgragmbj bg j`ug`aðh sj

bbglg J""N"

@ughkn uhg j`ug`aðh prjsjhtg snbn kjravgkgs nrkahgrags kj uhg n lgs

vgragmbjs kjpjhkajhtjs `nh rjspj`tn g uhg snbg vgragmbj ahkjpjhkajhtj

Jcjlpbns#

&&

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kx

vk 

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uk 

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ky

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$a bg %uh`aðh kjs`nhn`akg kjpjhkj kj l&s kj uhg vgragmbj bg j`ug`aðh sj

bbglg J""Y"

Pa uhg j`ug`aðh `nhtajhj kjravgkgs pgr`agbjs kj uhg n lçs vgragmbjs

kjpjhkajhtjs kj kns n lçs vgragmbjs ahkjpjhkajhtjs sj ka`j quj js uhg

j`ug`aðh ka%jrjh`agb pgr`agb 'JY(

Jcjlpbns#

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NrkjhNrkjh Jb   !kjh kj uhg j`ug`aðh kadjrjh`agb !nrkahgrag n jh kjravgkgs pgr`agbjs) js bg kjravgkg lçs gbtg `nhtjhakg jh jbbg

JcjlpbnKjtjrlahgr jb nrkjh kj bgs j`ug`anhjs kadjrjh`agbjs3

56   :0

∝

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FrgknFrgknJb frgkn kj uhg j`ug`aðh kadjrjh`agb js jb frgkn gbfjmrga`n kjsu kjravgkg kj lgnr nrkjh! js kj`ar! jb frgkn kj uhg

j`ug`aðh kadjrjh`agb js bg pntjh`ag g bg "uj jstg jbjvgkg bgkjravg "uj hns kan jb nrkjh kj bg j`ug`aðh kadjrjh`agb#JcjlpbnJb frgkn kj bg j`ug`aðh kadjrjh`agb js3

kj tjr`jr frgkn! kgkn "uj bg praljrg kjravgkg jst$ jbjvgkg

`umn#

56 :0

∝

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Kjtjrlahgr jb nrkjh kj bgs safuajhtjs j`ug`anhjs3

g%

 m%

60: =:

=

=

=

&

& +

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Kjtjrlahgr jb nrkjh frgkn kj bgs safuajhtjs j`ug`anhjs

kadjrjh`agbjs3

g% m%

`%

k%

ykx 

ky x 

kx 

yk :0

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%*)!  

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J@ZG@ANHJ$ ADJ!JH@AGBJ$ HN BAHJGBJ$

&hg j`ug`aðh kadjrjh`agb nrkahgrag hn bahjgb js salpbjljhtj uhg quj js hn

bahjgb. %gs duh`anhjs hn bahjgbjs kj bg vgragmbj kjpjhkajhtj n sus kjravgkgs,

`nln #jh (y) n , hn pujkjh gpgrj`jr jh uhg j`ug`aðh bahjgb pnr `nhsafuajhtj' y

j  ‽

 xj y y y  

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 KJP@]A$@ANH KJ BGP J"K KJP@]A$@ANH KJ BGP J"K""

Bg ahtjrprjtg`aðh kj uhg j`ug`aðh kadjrjh`agb js bg kjs`rap`aðh

lgtjlçta`g kj bg laslg pgrg jbbn sj lnstrgrg sjf,h su nrkjh+tapn y frgkn-

 

[apn Nrkjh "rgkn

Bahjgb

Nrkahgrag 7 7 s

  Ygr`agb

 x% y»»&xy»&y < 9 Nrkahgrag

 yy»»&x' y < x  Nrkahgrag #n/Yn!quj j `nj9`ajh&j  "j y 77 hn

"j(jh"j "j

 x j'`usavg*jh&j=.

   y»& y < xy Nrkahgrag #jh y»& y

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 xkx

ky`ns<

%&&

&

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DZH@ANHJP Y]ALA[ATGPDZH@ANHJP Y]ALA[ATGP%g duh`aðh y = 0(;, `#, `*, `", :, `h  ) kj h `nhstghtjs grmatrgrags sj kjhnlahg

duh`aðh pralatavg n pralatavg kj bg J`ug`aðh adjrjh`agb ! ( ;, y4, y44, :, y h) g

quajh sgtasdg`j. %g duh`aðh pralatavg gsa kjdahakg sj kjhnlahg tglma6h 5 dglabag kj

`urvgs ahtjfrgbjs1, pnr `nhsafuajhtj, bg J.. js bg j`ug`aðh kadjrjh`agb kj bgs

dglabags

J`ug`aðh a%jrjh`agb  Duh`aðh pralatavgD ' 97 -7 -= ( 0 ;D ' 97 -7 -== ( 0 ;D ' 97 -7 -==7-=== ( 0 ;D ' 97 -7 -= : -h ( 0 ; 

! ( ;, y, `# )

! ( ;, y, `#, `* )

! ( ;, y, `#, `*, `", )

! ( ;, y, `#, `*, :, `h )

 

HN*G# jb huljrn kj `nhstghtjs 4uj prjsjhtg bg %uh`aðh pralatavg js

a)ugb gb nrkjh kj su j`ug`aðh ka%jrjh`agb

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@NHN@AKG BG DZH@ANH Y]ALA[ATG 0GBBG] PZ J@ZG@ANH@NHN@AKG BG DZH@ANH Y]ALA[ATG 0GBBG] PZ J@ZG@ANHKADJ]JH@AGBKADJ]JH@AGB

Jb prnmbjlg js kjku`ar bg j`ug`aðh ka%jrjh`agb kj nrkjh h7 bamrj kj

`nhstghtjs grmatrgrags7 `nrrjspnhkajhtjs g uhg %uh`aðh pralatavg kj 6h5

`nhstghtjs grmatrgrags

J/J1YBN

$Z3J!JH@AG

(# )msjrvjlns jb h*ljrn kj `nhstghtjs kj ahtjfrg`aðh "uj gpgrj`jhjh bg +uh`aðh pralatavg kgkg#

=# ,a bg +uh`aðh pralatavg tajhjh h  `nhstghtjs kj ahtjfrg`aðh

kadjrjhtjs! jhtnh`js kjravgrjlns h vj`js tgb snbu`aðh

Y]NMBJLGP KJ TGBN] AHA@AGB ; TGBN] KJY]NMBJLGP KJ TGBN] AHA@AGB ; TGBN] KJ

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Y]NMBJLGP KJ TGBN] AHA@AGB ; TGBN] KJY]NMBJLGP KJ TGBN] AHA@AGB ; TGBN] KJD]NH[J]GD]NH[J]G

%)7!  

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 J/j7b3 $!=bj0 kj v0b! h;0b ;h v0!0# J/j7b3 $!=bj0 kj v0b! h;0b ;h v0!0# #bu;hj# #bu;hj#

=lmgs duh`anhjs y < 9 y  y < x: 46 sgtasdg`jh bg j`ug`aðhkadjrjh`agb kxky < xy'(% , y bg `nhka`aðh aha`agb y < 9, kjlnkn $uj jb prnmbjlg kjb vgbnr aha`agb

kxky < xy4(% ,  y(9)< 9

tajhj kns snbu`anhjs `ughkn ljhns. nln vjlns jh bgdafurg+ bg frgda`gs glmgs duh`anhjs pgsgh pnr jb lasln

 puhtn+ !%+ %)

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-hg snbu`aðh kj uhg JK js tnkg rjbg`aðh jhtrj bgs

 vgragmbjs "uj ahtjrvajhjh jh ka`.g j`ug`aðh "uj hn`nhtjhfgh hahfuhg kjravgkg "uj sgtasdgfg ak/hta`gljhtj gka`.g j`ug`aðhJcjlpbn

g duh`aðh kjdahakg pnr js uhg snbu`aðh kj bgj`ug`aðh kadjrjh`agb3

pujstn "uj3

  gb sustatuar jh bg JK sj nmtajhj uhg akjhtakgk

=

x1 y    ∝<

 y 

x y'   ∝<

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=(

=

7

=7

=

(

 x 

 x  x  x 

∝

∝

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PNBZ@A:H -JHJ]GB N @NLYBJ[GPNBZ@A:H -JHJ]GB N @NLYBJ[G

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-hg snbu`aðh pgrta`ubgr kj uhg JK js tnkg snbu`aðhnmtjhakg gsafhghkn vgbnrjs jspj`da`ns g bgs `nhstghtjs "ujahtjrvajhjh jh bg snbu`aðh fjhjrgb#

Jcjlpbn jrada`gr "uj y

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Kjravghkn y

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Kjtjrlahj sa `gkg j`ug`aðh js snbu`aðh n hn kj bg j`ug`aðhkadjrjh`agb kgkg3

(#

=#

0#

y x kx 

ky x @x  x y   +

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Jjr`a`ans gka`anhgbjs

Kjtjrlahj sa `gkg j`ug`aðh js snbu`aðh n hn kj bg j`ug`aðhkadjrjh`agb kgkg3(#

=#

0#

( ) =&(=   y x  xyy@x @ y  

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5!0yj;t!0# N!t1h0bj#5!0yj;t!0# N!t1h0bj#

 B0# t!0yj;t!0# !t1h0bj# snh bgs `urvgs $uj sjahtjrsj`tgh dnrlghkn uh çhfubn rj`tn.

Ygrg nmtjhjr bgs trgyj`tnrags nrtnfnhgbjs kj uhg

j`ug`aðh kadjrjh`agb+ sj tnlg- 4< , `nln

%<

%; kj bg trgyj`tnrag nrtnfnhgb g bg praljrg j`ug`aðh.

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 Jx#tjh;0 j uh;k0k Jx#tjh;0 j uh;k0k

=b rjsnbvjr uh prnmbjlg kj vgbnr aha`agb surfjh kns gsuhtns

duhkgljhtgbjs-

:46astj uhg snbu`aðh gb prnmbjlg 5+ sa bg 1gy :js ,ha`g5

Ygrg uh prnmbjlg kj vgbnr aha`agb + jh uhg j`ug`aðh+ sj prjfuhtg bn

safuajhtj- 

:Bg j`ug`aðh kadjrjh`agb tajhj

46astjh`ag snbu`anhjs 5  :=bfuhg `urvgs snbu`aðh pgsg pnr jb puhtn (x9,  y9 )? 

:uçhkn pnkjlns jstgr sjfurns $uj 1gy

3ha`akgk  prj`asgljhtj uhg `urvg snbu`aðh $uj pgsg pnr jb  puhtn (  x9,  y9 )?

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@07 k!j;;h0b @07 k!j;;h0b 

Bg tjrhg (x, y, y») kjtjrlahg bg karj``aðh kj uhgrj`tg $uj pgsg pnr jb puhtn (x, y)" 4b `nh?uhtn kj bnssjfljhtns kj jstgs rj`tgs js bg rjprjsjhtg`aðh

 1j2t!;0 kjb ;07 k!j;;h0b"

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Pj pujkj rjsnbvjr uhg j`ug`aðh kadjrjh`agb trg2ghkn jb

`glpn karj``anhgb+ jh knhkj+ pgrg `gkg `urvg kj bg

dglabag kj snbu`aðh+ bg tghfjhtj jh `gkg uhn kj sus

 puhtns tajhj bg laslg karj``aðh $uj jb `glpn jh jsj

 puhtn.

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MambanfrgdìgsMambanfrgdìgs

4`ug`anhjs jka`aðh 7I/(4k. "rupn 4katnragb Gmjrngl0ra`g

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4`ug`anhjs jka`aðh 7II&

4k. Y4=@PN# + =kkasnh Kjsbjy Bnflgh

3@B-1ttp-JJmnnDs.fnnfbj.`nl.l6JmnnDs5ak;dF=8ljGY%m=Epf;Y=&E

k$;AujCjsCjbCfrgknCjhCuhgCj`ug`anhCkadjrjh`agbjsE` 

http://books.google.com.mx/books?id=fKAZmeIP0bAC&pg=PA23&dq=Que+es+el+grado+en+una+ecuacion+diferenciales&chttp://books.google.com.mx/books?id=fKAZmeIP0bAC&pg=PA23&dq=Que+es+el+grado+en+una+ecuacion+diferenciales&chttp://books.google.com.mx/books?id=fKAZmeIP0bAC&pg=PA23&dq=Que+es+el+grado+en+una+ecuacion+diferenciales&chttp://books.google.com.mx/books?id=fKAZmeIP0bAC&pg=PA23&dq=Que+es+el+grado+en+una+ecuacion+diferenciales&c

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"bnsgran"bnsgran

Tgragmbjs kjpjhkajhtjs.O Pnh g$ujbbgs vgragmbjs$uj `nln su hnlmrj bn ahka`g+ kjpjhkjh kjb vgbnr

$uj tnlg bgs ntrgs vgragmbjs Ynr j?jlpbn-  .(x)< x+ y n .(x) js bg vgragmbj kjpjhkajhtj yg $uj jstg su?jtg g

bns vgbnrjs $uj sj bj sumlahastrj g x.

Tgragmbj Ahkjpjhkajhtj.O 4s g$ujbbg vgragmbj $uj

hn kjpjhkj kj hahfuhg ntrg vgragmbj+ jh jb j?jlpbn

ghtjranr bg x js bg vgragmbj ahkjpjhkajhtj yg $uj bg yjs bg $uj kjpjhkj kj bns vgbnrjs kj x" 

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""

J.K.N "j vg!ag$js sj(g!g$jsJ.K.N "j vg!ag$js sj(g!g$js

  &hg j`ug`aðh ka%jrjh`agb kj vgragmbjs sjpgrgmbjs js uhg j`ug`aðh

kadjrjh`agb kj praljr nrkjh jh bg `ugb bg j;prjsaðh pgrg kykx  sjpujkj dg`tnra3gr `nln'

( ) ( ) y 1  x . kxky <n tglma6h `nln'

( )( ) y 1 

 x . 

kx

ky< sa 1 ( y)  7.≩

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Js`ramalns bg j`ug`aðh sjpgrgmbj jh dnrlg kadjrjh`agb'

( )

  ( )kx x . ky

 y 1 

<7

n tglma6h `nln'

( ) ( )kx x . ky y 1 

  <sjf?h sjg jb `gsn. %ujfn ahtjfrglns, `nh rjspj`tn g y jh jb

lajlmrn a3quajrkn y `nh rjspj`tn g x jh jb lajlmrn kjrj`2n.

sa 1 ( y)  7.≩

]JPNBZ@ANH KJ J.K.N. @NH TG]AGMBJ]JPNBZ@ANH KJ J.K.N. @NH TG]AGMBJPJYG]GMBJPJYG]GMBJ