9 differential equations
TRANSCRIPT
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@ H C C MT M A P H K J M [ ] K P H G AV 5,x
xyx
`x`y 44 -+> gr
xy
`x`y > +
xy
sha , mtf, h.m,
fgndham` `mlrmm gc mkfb tmrn havgjvm` hs sknm gr
>
zy
`x`y
c . \m kjsg sky tbkt tbm
bgnglmamgus cuafthga havgjvm` hs gc `mlrmm zmrg. Ka mqukthga hs k bgnglmamgusmqukthga, hc cgr cuafthga havgjvm` c ( j x, j y) > c (x, y).
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Ha sufb fksms, wm sudsthtutm`x`v
xv`x`y
vxygr vxy +>>> ka` prgfmm` dy sudsthtuthal cgr
`x`y
ka` y ha tbm lhvma mqukthga. Vgnmthnms, wm nhlbt lmt tbm bgnglmamgus mqukthga ks
>
yx
`y`x c . Ha sufb fksm, wm sudsthtutm x > vy `y
`x > v + `y
`vy ka` prgfmm` dy sudsthtuthal
cgr `y`x
ka` x ha tbm lhvma mqukthga.
(f) Ha jhamkr `hccmrmathkj mqukthga gc chrst gr`mr )x(U`x`y + . y > [(x) 9 U(x), [(x) krm
cuafthga gc x.
Cha` Hatmlrkthal Ckftgr (H.C.) > `xUm ka usm tbm rmsujt y. (H.C.) > )}x([}C.H{( x ks sgjuthga.
Vgnmthnms, wm lmt jhamkr hccmramthkj mqukthga cgrn ks )y([),y(U)9y([x).y(U`y`x >+ krm cuafhtgas
gc y. Ha sufb fksms cha` H.C. > `yUm ka` usm tbm rmsujt x. (H.C.) > `y)}y([.)C.H{( ks sgjuthga.
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8/10/2019 9 Differential Equations
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VGJSM@ UTGDJMVN
Mx.; \rhtm tbm gr`mr ka` mlrmm gc tbm hccmrmathkj mqukthga 5`x`y
sha`x`y >
Vgj . Bhlbmst gr mr mrhvkthvm hs`x
`y
Z Gr`mr gc hccmrmathkj mqukthga hs ;.Mqukthga fkagt dm wrhttma ks k pjyagnhkj ha mrhvkthvms. Bmafm mlrm hs agt mcham`.
Mx.4 \bkt whjj dm tbm gr mr gc tbm hccmrmathkj mquktha, fgrrmspga`hal tg tbm cknhjy gc furvmsy > k sha (x + d), wbmrm k hs krdhtrkry fgastkat.
Vgj. Ks tbmrm hs gam krdhtrkry fgastkat, sg gr`mr gc fgrrmspga`hal `hccmrmathkj mqukthga hs ;.
Mx.6 Cgrn tbm hccmrmat mqukthga rmprmsmathal tbm cknhjy gc furvms lhvma dy xy > Km x + Dm x wbmrm K ka` D krm fgastkats.
Vgj. xy > Kmx
+ Dmx
...(h)y + xy > Km x Dmx ka` y + y + xy" > xy YCrgn(h)^ xy' + 4y xy > 5 hs tbm rmquhrm`mqukthga
Mx.0 Cha` tbm mqukthga gc k furm pksshal tbrgulb tbm pghat (4, 6), lhvma tbkt tbm sjgpm gc tbm
tkalmat tg tbm furvm kt kay pghat (x, y) hs 4yx4
.
Vgj. \m iagw tbkt tbm sjgpm gc tbm tkalmat tg k furvm kt kay pghat (x, y) hs lhvma dy`x`y
.
Vg, 4yx4
`x`y > Hatmlrkthal dgtb sh`ms, wm bkvm
> `xx4`yy 4 Fx6y 4
6
+> ...(;)
Vhafm tbm furvm (;) pkssms tbrgulb tbm pghat (4, 6) F)4(6)6( 4
6
+-> 8 > 0 + F F > =
Vudsthtuthal tbhs vkjum gc F ha (;), wm bkvm =x6y 4
6
+> h.m., y6
6x4
> ;= gr y >6 4
;=x6 +
Mx.= Kt kay pghat (x, y) gc k furvm, tbm sjgpm gc tbm tkalmat hs twhfm tbm sjgpm gc tbm jham smlnmat oghahal tbm pghat gc fgatkft tg tbm pghat (0, 6). Cha` tbm mqukthga gc tbmfurvm, lhvma tbkt ht pkssms tbrgulb (4, ;).
Vgj. Ht hs lhvma tbkt
++>
---->
x0y6
4x0y6
4`x`y
`xx0
4`y
y6;
+>
+
Ga hatmlrkthal, wm bkvm, jgl |6 + y| > 4 jgl | 0 + x | + FVhafm tbm furvm pkssms tbrgulb tbm pghat (4, ;)
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8/10/2019 9 Differential Equations
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wm sudsthtutm tbm vkjum x > 4 ka` y > ; ha Mq. (;) ka` lmtjgl |6 + ;| > 4 jgl |0 4| + F
F > jgl 0 4 jgl 4 > 4 jgl 4 4 jgl 4 > 5Bmafm, crgn (;), tbm rmquhrm` mqukthga gc tbm furvm hsjgl |6 + y| > 4 jgl |0 + x| gr 6 + y > (0 + x) 4 h.m., y + 6 > (x + 0) 4
Mx.3 K pgpujkthga lrgws kt tbm rktm gc 4% pmr ymkr. Bgw jgal gms ht tkim cgr tbm pgjujkthga`g gudjm 7
Vgj. Jmt U5 dm tbm hahthkj pgpujkthga ka` jmt tbm pgpujkthga kctmr t ymkrs dm U. Pbma,
`U 4U`t ;55
> (lhvma)
`t=5;
U`U > Hatmlrkthal dgtb sh`ms, wm bkvm
jgl U > Ft=5
; + \bma t > 5, U > U5
Z jgl U5 > F Bmafm, t
=5
;
U
Ujgl
5
>
\bma U > 4U 5
>
5
5
UU4
jgl=5t > =5 jgl 4 > =5 5.6 > ;= ymkrs
Pbus, ht tkims ;= ymkrs cgr tbm pgpujkthga tg dm `gudjm.
Mx.< Vbgw tbkt tbm hccmrmathkj mqukthga xxy
fgsy`x`y
xy
fgsx
>
hs bgnglmamgus ka sgjvm
ht.Vgj. Pbm lhvma hccmrmathkj mqukthga fka dm wrhttma ks
+
>
xyfgsx
xxyfgsy
`x`y
...(;)
Ht hs k hccmrmathkj mqukthga gc tbm cgrn )y,x(l)y,x(c
`x`y >
Bmrm, c(x,y) > xxy
fgsy +
ka` l(x, y) >
xy
fgsx Mkfb gam hs fjmkrjy k bgnglmamgus cuafthga gc
`mlrmm ;.
Uut y > a x gr a>xy
...(4) @hccmrmathkthal mqukthga (4) whtb rmspmft tg x, wm lmt
`x`
x`x`y a+a> ...(6)
Vudsthtuthal tbm vkjum gc y ka` `x`y
ha mqukthga (;), wm lmt a+aa
>a
+a fgs ;fgs`x`x
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8/10/2019 9 Differential Equations
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gr a-a+aa>aa
fgs;fgs
`x`
gr a>a
fgs;
`x`
x ...(0)
Vmpkrthal tbm vkrhkdjms ha mqukthga (0), wm lmt fgs a ` a >x
`x...(=)
Hatmlrkthal dgtb sh`ms gc mqukthga (=), wm lmt
>aa `xx;
`fgs
gr sha a > jgl |x| + jgl |F|gr sha a > jgl |Fx| ...(3)
Tmpjkfhal a dyxy
ha mqukthga (3), wm lmt |Fx|jglxy
sha >
gr y > x sha ; {jgl |Fx|} ...(
Vgj. Pbm hccmrmathkj mqukthga fka dm wrhttma ks
xxy
shaxy
`x`y
-
>
c (x, y) hs k bgnglmamgus cuafthga gc `mlrmm ;.bgnglmamgus cuafthga gc gr`mr ;.Pbus, tbm lhvma mqukthga hs k bgnglmamgus hccmrmathkj mqukthga.
Uut y > ax sg tbkt`x`
x`x`y a+a>
Vudsthtuthal tbm vkums gc y ka``x`y
ha (;), wm bkvm
a-a>
a-a
>a
+a shaxxx
shaxx
`x`
x a>
asha
`x
`x
x
`x`.
sha
; ->aa
Hatmlrkthal dgtb sh`ms, wm bkvm
->aa x`x
`sha
; +->aa |Fjgl|xjgl`mffgs |Fjgl|xjgl4tkatka +->
a
4
tka a
>xF
xF
x4y
tka > h.m., Fx4
ytkax >
Mx.8 Vbgw tbkt tbm mqukthga gc tbm furvm wbgsm sjgpm kt kay pghat (x, y) hs mqukj tg y + 4xka` wbhfb pkssms tbrgulb tbm grhlha dy y + 4(x + ;) > 4m x
Vgj. Bmrm, wm bkvm
x4y`x`y +> x4y
`x`y >-
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Bmrm, U > ; ka [ > 4x
Z H.C. > -> `x.;;`xU
mm x
x
m;
m >> -
Vg, tbm lmamrkj sgjuthga gc tbm hccmrmathkj mqukthga (;) hs
+> F`xmx4m;.y xx +> - F`xm.x4m;.y xx F^`xmxmY4 xx ++-> -- ymx > 4xm x 4m x + FVhafm tbm furvm pkssms tbrgulb tbm grhlha (5, 5) wm bkvm
Fm4m.5.4m5 555 +--> --- F > 4
Bmafm, tbm mqukthga gc tbm furvm hsymx > 4xm x 4m x + 4 h.m., y > 4x 4 + 4m x gr y + 4(x + ;) > 4m x
Mx.;5 Cha` tbm mqukthga gc k furvm pksshal tbrgulb tbm pghat (5, 4) lhvma tbkt tbm sun gc tbmfggr`haktms gc kay pghat ga tbm furvm mxfmm`s tbm nklahtu`m gc tbm sjgpm gc tbm tka-lmat tg tbm furvm kt tbkt pghat dy =.
Vgj. Kffgr`hal tg tbm qumsthga,
x + y > =`x`y +
`x`y
y > x =
Bmrm, U > ; ka` [ > x = ...(;)
x`x;`xU mmm.C.H --
>>> Vg, tbm lmamrkj sgjuthga gc tbm @.M. (;) hs
+-> -- F`xm)=x(ym xx +-> --- F`xm=`xxmym xxx
Fm=mxmym xxxx ++--> ---- y > x ; + = + Fm x
Vhafm tbm furvm pkssms tbgrulb (5, 4), 4 > 5 ; + = + Fm 5 F > 4
Mx.;; Cha` tbm mqukthga gc k furvm pksshal tbrgulb tbm pghat (5, ;). Hc tbm sjgpm gc tbm tka-lmat tg tbm furvm kt kay pghat (x, y) hs mqukj tg tbm sun gc tbm x-fggr`haktm (kdsfhssk)ka` tbm prg`uft gc tbm x-fggr`haktm ka` y - fggr`haktm (gr`haktm) gc tbkt pghat.
Vgj. Kffgr`hal tg tbm qumsthga,
xyx`x`y +> xxy
`x`y >- ............(;)
Bmrm, U > x ka [ > x 4x
x`x`xU4
mmm.C.H
-->>>
Vg, tbm lmamrkj sgjuthga gc tbm @.M. (;) hs +> -- F`xxmym 4x
4x 44
............(4)
Jmt -
> `xxmH 4x4
Uut t4x 4 >- sg tbkt x `x > `t Z
-->->-> 4
xtt
4
mm`tmH
Bmafm, crgn (4) Fmym 4x
4x 44
+-> --
h.m., 4x 4
Fm;y +->
Vhafm tbm furvm (6) pkssms tbrgulb (5, ;), ............(6)
; > ; + Fm 5 F > 4
Vg, tbm rmquhrm` mqukthga gc tbm furvm hs 4x4
m4;y >+
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8/10/2019 9 Differential Equations
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@HCCMTMAPHKJ M[]KPHGAV 5 (hh) x
44
4 m0y8
`xy`
x; -->+
(hhh) 4 4
4`y ; yxy (; x x )`x ; x
+> + + +
[.4 Cgrn tbm hccmrmathkj mqukthga fgrrmspga`hal tg y 4 4ky + x 4 > k 4 dy mjhnhakthal k.
[.6 Cgrn tbm hccmrmathkj mqukthga gc tbm mqukthga (x + k) 4 4y 4 > k 4 dy mjhnhakthal k.
[.0 Cgrn tbm hccmrmathkj mqukthga fgrrmspga`hal tg (x k) 4 + 4y 4 > k 4 dy mjhnhakthal k.
[.= Cgrn tbm hccmrmathkj mqukthga y > kx 4 + dx + f. mjhnhakthal k, d, f.
[.3 Vbgw tbkt y > K sha x + D fgs mx + x sha x hs k sgjuthga gc tbm hccmrmathkj mqukthga y + 44
`x
y` >
4m fgs x
[.< Vbgw tbkt y > jgl (x + 44 kx + ) 4 skthschms tbm hccmrmathkj mqukthga (k 4 + x 4) 44
`x
y` + x
`x`y
> 5
[.2 Vbgw tbkt y > xtka ;
fm-
hs k sgjuthga gc tbm hccmrmathkj mqukthga (; + x) 4 44
`x
y` + (4x ;)
`x`y
> 5
[.8 Vbgw tbkt y > Kx +xD
, x 5 hs k sgjuthga gc tbm `hccmrmathkj mqukthga 5y`x`y
x`x
y`x
44 >-+
[.;5 Vgjvm tbm hccmrmathkj mqukthga`x`y
> fgs 6 x sha 4 x + x ;x4 + , x
- ,
4;
[.;; Vgjvm tbm hccmrmathkj mqukthga fgs x`x`y
fgs 4x > fgs 6x
[.;4 Vgjvm tbm hccmrmathkj mqukthga (; + x 4)`x`y
x > 4 tka ; x
[.;6 Vgjvm tbm `hccmrmathkj mqukthga fgs y `y + fgs x sha y `x > 5 lhvma tbkt y > 4p
wbma x > 4p
[.;0 Vgjvm tbm hccmrmathkj mqukthga >`x`y
m x + ;, lhvma tbkt y > 6 wbma x > 5.
[.;= Vgjvm tbm hccmrmathkj mqukthga (; + y 4) `x xy `y > 5, ka` wbhfb pkssms tbrgulb (;, 5).
[.;3 Vgjvm tbm hccmrmathkj mqukthga x`x`y
sha
xy
+ x y sha
xy
> 5, y (;) >4p
[.;< Vgjvm tbm hccmrmathkj mqukthga xm y/x y sha4p + x
`x`y sha xy > 5, y(;) > 5
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[.;2 Vgjvm tbm hccmrmathkj mqukthga xy jgl yx
`x +
-
yx
jglxy 44 `y > 5, lhvma tbkt y (;) > m
[.;8 Vgjvm tbm hccmrmathkj mqukthga`x`y
+ y fgt x > 4x + x 4 fgt x (x 5) lhvma tbkt y ( p /4) > 5
[.45 Vgjvm tbm `hccmrmathkj mqukthga (; + y 4) x > (tka ; y x)`x, y(5) > 5
[.4; Vgjvm tbm hccmrmathkj mqukthga x`x`y
y > jgl x, y(;) > 5
[.44 Pbm sjgpm g tbm tkalmat kt (x, y) tg k furvm pksshal tbrgulb
p
0,; hs lhvma dy
xy
fgs 4xy
.
Cha` tbm mqukthga gc tbm furvm.
[.46 Pbm surckfm krmk gc k dkjjgga dmhal hacjktm` fbkalms kt k rktm prgpgrthgakj tg thnm t. Hc hahthkjjy hts rk`hus hs ; uaht ka` kctmr ; smfga` ht hs 6 uahts, cha` tbm rk`hus kctmr t smfga`s.
[.40 Pbm pgpjukthga lrgws kt tbm rktm gc =% pmr ymkr. Bgw jgal gms ht tkim cgur tbm pgpujkthga tg`udjm 7
[.4= K rk`hgkfthvm sudstkafm `hshatmlrktms kt k rktm prgpgrthgakj tg tbm knguat gc sudstkafmprmsmat. Hc =5% gc tbm lhvma knguat `hshatmlrktms ha ;355 ymkrs. \bkt prmfmatklm gc tbmsudstkafm hshatmlrktms ha ;5 ymkrs 7 YPkim m jgl 4/;35 > 5.88=
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DGKT@ UTGDJMV
MRMTFHVM HH
[.; Vgjvm tbm hccmrmathkj mqukthga`x`y
x > y(jgl y jgl x ;) YF.D.V.M. 455;^
[.4 Vgjvm tbm hccmrmathkj mqukthga xyyx;`x`y -+-> YF.D.V.M. 4554^
[.6 Vgjvm tbm hccmrmathkj mqukthga (x 6 + y 6)`y x 4y`x > 5 YF.D.V.M. 4554^
[.0 Vgjvm tbm hccmrmathkj mqukthga (y + xy)`x + (x xy 4) `y > 5 YF.D.V.M. 4554^
[.= Vbgw tbkt tbm hccmrmathkj mqukthga gc y 4 > 0k(x d) hs 5`x`y
`xy`
y4
4
4
>
+ YF.D.V.M. 4556^
[.3 Vgjvm tbm hccmrmathkj mqukthga6xy
`x
`yx >+
wbma y(4) > ; YF.D.V.M. 4556^
[.< Cha` tbm hccmrmathkj mqukthga gc tbm cknhjy gc furvms lhvma dy x 4 + y 4 > 4kx. YF.D.V.M. 4556^
[.2 Vgjvm tbm hccmrmathkj mqukthga xtkay`x`y
xfgs 4 >+ YF.D.V.M. 4556^
[.8 Cgrn tbm hccmrmathkj mqukthga gc xy > Km x + Dm x + x 4 wbmrm K ka` D krm fgastkatYF.D.V.M. 4556^
[.;5 Cgrn tbm hccmrmathkj mqukthga gc furvm y 4 4ky + x 4 > k 4, wbmrm k hs ka krdhtrkry fgastkat.YF.D.V.M. 4550^
[.;; Vgjvm tbm hccmrmathkj mqukthga xshaxtkay4`x`y >+ YF.D.V.M. 4550^
[.;4 Vgjvm tbm hccmrmathkj mqukthga (x 4 yx 4) y + (y 4 + x 4y 4) `x > 5. YF.D.V.M. 4550^
[.;6 Vgjvm tbm hccmrmathkj mqukthga 5yxy4`x`y
x4 44 >+- wbma y(m) > m. YF.D.V.M. 455=^
[.;0 Vgjvm tbm hccmrmathkj mqukthga 5x,mxy
`x`y x 1>+ YF.D.V.M. 455=^
[.;= Vgjvm tbm hccmrmathkj mqukthga (x 4 + xy) y + (x 4 + y 4) x YF.D.V.M. 455=^
[.;3 Vgjvm tbm hccmrmathkj mqukthga (; + x 4) xy4`x`y - > (x 4 + 4) (x 4 + ;) YF.D.V.M. 455=^
[.;< Vgjvm tbm hccmrmathkj mqukthga`x`y
+ smfx. y > tka x YF.D.V.M. 4553^
[.;2 Vgjvm tbm hccmrmathkj mqukthga 4xy x + (x 4 + 4y 4) y > 5 YF.D.V.M. 4553^
[.;8 Cgrn tbm hccmrmathkj mqukthga gc y > k sha (x + d), wbmrm k ka` d krm krdhtrkry fgaskats.
[.45 Cgrn tbm hccmrmathkj mqukthga gc y > K fgs 4x + D sha 4x wbma K, D krm fgastkatYF.D.V.M. 455
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[.4; Vgjvm tbm hccmrmathkj mqukthga (x fgs y)`y > m x(x jgl x + ;)`x YF.D.V.M. 455 sha x YF.D.V.M. 4552^
[.46 Vgjvm tbm hccmrmathkj mqukthga x 4 `x`y
+ 4y tka x > sha x YF.D.V.M. 4552^
[.40 Cgrn tbm hccmrmathkj mqukthga cgr tbm fhrfjms pksshal tbrgulb grhlha ka` whtb fmatrms ga y-kxhs. YF.D.V.M. 4552^
[.4= Vgjvm tbm hccmrmathkj mqukthga x`x`y
> y x tkaxy
YF.D.V.M. 4558^
[.43 Vgjvm tbm hccmrmathkj mqukthga fgs 4x`x`y
+ y > tka x YF.D.V.M. 4558^
[.4< Vgjvm tbm hccmrmathkj mqukthga`x`y
+ y > fgs x sha x YF.D.V.M. 4558^
[.42 Vgjvm tbm hccmrmathkj mqukthga (x 4 + ;)`x`y
+ 4xy > 0x4 + YF.D.V.M. 45;5^
[.48 Vgjvm tbm hccmrmathkj mqukthga (x 6 + x 4 + x + ;)`x`y
> 4x 4 + x YF.D.V.M. 45;5^
[.65 Vgjvm tbm hccmrmathkj mqukthga x y y x > 44 yx + x YF.D.V.M. 45;;^
[.6; Vgjvm tbm hccmrmathkj mqukthga (y + 6x4
) `y
`x
> x YF.D.V.M. 45;;^
[.64 Cgrn tbm hccmrmathkj mqukthga gc tbm cknhjy gc fhrfjms ha tbm smfga`. quk`rkat ka` tgufbhaltbm fgg`haktm kxms. YF.D.V.M. 45;4^
[.66 Cha` tbm pkrthfujkr sgjuthga gc tbm hccmrmathkj mqukthga x(x 4 ;) `x`y > ;9 y > 5 wbma x > 4.
YF.D.V.M. 45;4^[.60 Vgjvm tbm cgjjgwhal hccmrmathkj mqukthga ?
YF.D.V.M. 45;4^(e; + x 4) `y + 4 xy `x > fgt x `x9 x 5
[.6= Cha` tbm pkrthfujkr sgjuthga gc tbm hccmrmathkj mqukthga YF.D.V.M. 45;6^(tka ; y x) `y > (; + y 4) `x, lhvma tbkt wbma x > 5, y > 5.
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KAV\MT IMXMRMTFHVM ; (]AVGJSM@ UTGDJMNV)
;. (h) gr`mr-;, mlrmm-; (hh) gr`mr-4, mlrmm-; (hhh) gr`mr-;, mlrmm-;
4. (x 4 4y 4)4
`x`y
0xy`x`y
x 4 > 5 6. x4 + 4y 4 > 0xy`x`y
0. 0xy`x`y
> 4y 4 x 4
=. 66
`x
y` > 5 ;5. y >
6;
sha 6 x =;
sha=x + ,F);x4(3;);x4(
;5; 4
64=
++-+ x
- ,
4;
;;. y > sha 4x x + 4 sha x + jgl |smf x + tka x| + F, x (4a + ;)4p
, a _
;4. y >4;
jgl |; + x 4| + (tka ; x) 4 + F ;6. jgl sha y + sha x > ; ;0. y > (x + ;) jgl |x + ;| x + 6
;=. x 4 > (; + y 4) ;3. jgl | x | > fgs
xy , x 5 ; +
;2. 44
y4
xjgl y
x 4
4
y0
x jgl y > ; 4m0
6;8. y > x 4
xsha0
4p (sha x 5) 45. (x tka ; y + ;)m tka ; y > ;
4;. y > x ; jgl x, x 1 5 44. y > x tka ;
xmjgl 46. ;t2 4 +
40. 45(jgl 4) ymkrs 4=. 5.06%
MRMTFHVM 4 (DGKT@ UTGDJMNV)
;. y > xm fx 4. jgl(; + y) > x 4
x4 + f 6. 66
y6
x- + jgl y > f 0. jgl x + x + jgl y
4y4 > f
=. xy >0
x 0 4 5 2. y > tka x ; + fm tka x 8. x 44
`x
y` + 4
`x`y
> xy x 4 + 4
;5. x4
- ;
`x`y
4y 44
`x`y
4xy`x`y
> 5 ;;. y > fgs x + f fgs 4x ;4. jgl y + y;
+x;
x > f
;6. y > xjgl;x4
+ ;0. y > x;
(x ;)m x +xf
;=. jgl x + f > 4 jgl
-
xy
; xy
;3. y > (x 4 + ;)(x + tka ; x) + f(x 4 + ;) ; smf x + tka x
;2. 6x 4y + 4y 6 > f ;8. 44
`xy` + y > 5 45. 4
4
`xy` + 0y > 5 4;. sha y > m x jgl x + f
-
8/10/2019 9 Differential Equations
12/12
@HCCMTMAPHKJ M[]KPHGAV;4
www.tbhaihht.ha
44. y > fgs x + fgs 4 x 46. y >4;
x (x + y) 40. (x 4 y 4)`x`y
> 4xy
4=. jgl fxy
xshax >
43. y > (tka x ;) + fm tka x 65. y + 44 yx + > fx4
6;. y > 6x 4 + fx 64. (x + y) 4 |(y) 4 + ;| > (x + yy) 4
66. 06jgl
4;
x;;jgl
4;y 4 -
-> 60. y > (; + x 4) ; jgl |sha x | + f (; + x 4) ;