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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 $uj hn `ulpbjh bgs prnpajkgkjs
ghtjranrjs.
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%&
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<∍∍
+∍∍
y
u
x
u
Jh bg lgynrìg kj bns bamrns bgs kjravgkgs nrkahgrags sj js`ramjh `nh bg
hntg`aðh kj Bjamhaz, n majh, `nh bg hntg`aðh kj Yralg
xj ykxky
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J@ZG@ANH ADJ!JH@AGB N!AHG!AG
Pa bg duh`aðh kjs`nhn`akg kjpjhkj kj snbn uhg vgragmbj bg j`ug`aðh sj
bbglg J""N"
@ughkn uhg j`ug`aðh prjsjhtg snbn kjravgkgs nrkahgrags kj uhg n lgs
vgragmbjs kjpjhkajhtjs `nh rjspj`tn g uhg snbg vgragmbj ahkjpjhkajhtj
Jcjlpbns#
&&
&
&
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+∝< kxku
kx
vk
kx
uk
xj ykx
ky
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$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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&
&
&
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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
0
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8/17/2019 Ecuaciones Diferenciales (UNSM-Tarapoto)-_basico
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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' ∝<
( ) ( )=
=(
=
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 $uj jb `glpn jh jsj
puhtn.
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MambanfrgdìgsMambanfrgdìgs
4`ug`anhjs jka`aðh 7I/(4k. "rupn 4katnragb Gmjrngl0ra`g
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4`ug`anhjs jka`aðh 7II&
4k. Y4=@PN# + =kkasnh Kjsbjy Bnflgh
[email protected].`nl.l6JmnnDs5ak;dF=8ljGY%m=Epf;Y=&E
k$;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