9 Differential Equations

8/10/2019 9 Differential Equations

1/12

@HCCMTMAPHKJ M[]KPHGAV;

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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).


2/12

@HCCMTMAPHKJ M[]KPHGAV4

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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.


3/12


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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, ;)


4/12


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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


5/12

@HCCMTMAPHKJ M[]KPHGAV=

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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 >-


6/12


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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 >+


7/12

@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


8/12


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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=


9/12


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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


10/12

@HCCMTMAPHKJ M[]KPHGAV;5

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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.


11/12

@HCCMTMAPHKJ M[]KPHGAV;;

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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


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) ;

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9 Differential Equations