Ukulinganisa okuhlukileyo okulandelelana kokuqala. Izisombululo zesampula

Ndicinga ukuba sifanele siqale ngembali yesixhobo esinokubaluleka semathematika njengendlela yokulinganisa. Njengazo zonke i-calculi ezahlukahlukileyo, ezilinganayo zenziwe nguNewton ekupheleni kwekhulu le-17. Wayejonga ukuba oku kufumanisa ukuba kubaluleke kakhulu kangangokuthi wadibanisa umyalezo, onokuthi uguqulelwe namhlanje: "Yonke imithetho yemvelo ichazwa ngokulinganisa." Kubonakala ngathi ukunyaniseka, kodwa kuyinyaniso. Nayiphi na umthetho we-physics, i-chemistry, i-biology inokuchazwa ngala mabalo.

Igalelo elikhulu ekuphuhliseni nasekudalweni kwembono yokulinganisa ngokwahlukileyo yenziwe ngabafundi beemathematika u-Euler noLagrange. Sekuqaleni kwekhulu le-18, bafumanisa kwaye bahlakulela oko kufundiswa ngoku kwiikholeji eziphezulu.

Olona nto ibalulekileyo ekufundeni ukulingana okuhlukileyo kwaqala ngoHenri Poincare. Wadala "inkolelo yokulinganisa imilinganiselo eyahlukileyo", edibanisa nenkolelo yemisebenzi yesigxina esiyinkimbinkimbi yenza inxaxheba ebalulekileyo kwisiseko sobukhosi - isayensi yendawo kunye nempahla yayo.

Ziziphi izinto ezihlukeneyo?

Abantu abaninzi banokwesibini ibinzana elithi "ukulinganisa ngokulinganayo". Nangona kunjalo, kweli nqaku, siya kubonisa yonke into ebalulekileyo yilezi zixhobo zethematika ezixhamlekileyo, eqinisweni ayinzima njengoko zibonakala kwisihloko. Ukuze uqale uxelele ngokulinganisa okuhlukileyo kokuqala, kufuneka uqale uqhelane neengcamango ezisisiseko ezihambelana nolu ngcaciso. Kwaye siza kuqalwa ngokwahlukileyo.

Ukwahlukana

Abantu abaninzi bayazi le ngcamango esikolweni. Nangona kunjalo, siya kuhlala kuyo ngokubanzi. Khawucinge ngomsebenzi wegrafu. Sinokuyandisa ukwenzela ukuba nayiphi na amacandelo ayo athathe ifom yomgca ochanekileyo. Kulo sithatha amaphuzu amabini asondelelene. Umehluko kwiinqununu zabo (x okanye y) ayingenamlinganiselo. Ibizwa ngokuba yohlukwano kwaye ibonakaliswe yimpawu (d) yohluko y) kunye ne-dx (ukuhlukana kwe x). Kubaluleke kakhulu ukuqonda ukuba ukuhlukahlula akusiyo ubuninzi obugqityiweyo, kwaye le nto inentsingiselo kunye nomsebenzi oyisiseko.

Kwaye ngoku kufuneka sicinge ngento elandelayo, esiyidingayo xa sicacisa ingcamango yokulinganisa. Oku kuyimvelaphi.

Isiphumo

Sonke sivakala ukuba besiva esikolweni kunye nale ngcamango. Kuthiwa ukuba i-derivative iyinqanaba lokukhula okanye ukuncipha komsebenzi. Nangona kunjalo, ininzi yale nkcazo ayinakuqondakala. Makhe sizame ukucacisa i-derivative ngokuhlukileyo. Masibuyele kwisiqwenga esingapheliyo somsebenzi ngamanqaku amabini, asemgama omncinci omnye komnye. Kodwa nangenxa yalo mda umsebenzi unalo ixesha lokutshintsha kwinqanaba elithile. Kwaye ukuchaza lo tshintsho kwaye ufike nge-derivative engabhalwa ngenye indlela njengomlinganiselo wohlukeneyo: f (x) '= df / dx.

Ngoku sifuna ukuqwalasela iipropati ezisisiseko ze-derivative. Kukho ezintathu kuphela:

1. Isiphumo semali okanye umahluko unokumelwa njengesixa okanye umahluko phakathi kweziphumo: (a + b) '= a' + b 'kunye (ab)' = a'-b '.
2. Ipropati yesibini ihambelana nokuphindaphinda. Imveliso evela kwixabiso lemveliso yomsebenzi omnye kwi-derived of another: (a * b) '= a' b + a * b '.
3. Imvelaphi yohlukwano ingabhalwa ngolu hlobo lulandelayo: (a / b) '= (' * b * b ') / b ² .

Zonke ezi zakhiwo zincedo ekufumaneni izisombululo zokulinganisa okokuqala ukulandelana.

Kukho iziphumo ezikhethiweyo. Masithi sinomsebenzi o oxhomekeke kwimimiselo x kunye y. Ukubala isiphumo esithile salo msebenzi, zithi, ngokubhekiselele ku-x, kufuneka sithathe iinguquguquliyo njengoko zihlala zize zihluke.

Udidi

Enye ingundoqo ebalulekileyo ibalulekileyo. Enyanisweni, oku kuhambelana ngokuthe ngqo kwinqununu. Ukuhlanganiswa kweentlobo eziliqela, kodwa ekuxazululeni ukulinganisa okulula kunokuba sifuna ukuhlanganiswa okungafani nakakhulu.

Ngoko, yintoni ebalulekileyo? Masithi sinexhomekeke ethile kwi f x. Sithatha kuwo udidi kwaye sifumane umsebenzi F (x) (odla ngokubizwa ngokuba yi-antiderivative), isisiseko esilingana nomsebenzi wokuqala. Ngaloo ndlela, F (x) '= f (x). Kwakhona kulandela ukuba udidi oluvela kwi-derivative lilingana nomsebenzi wokuqala.

Xa kusombulula ukulinganisa ngokulinganayo, kubaluleke kakhulu ukuqonda intsingiselo kunye nomsebenzi wokubambisana, kuba kubaluleke kakhulu ukuba kubamkele ukufumana isisombululo.

Ukulingana kuyahluka ngokuxhomekeka kwimeko yabo. Kwinqanaba elilandelayo, siza kuqwalasela iintlobo zokulinganisa kokuqala kokuhlukana komyalelo, uze ufunde indlela yokuzixazulula ngayo.

Iiklasi zokulinganisa ngokwahlukileyo

"I-diffusers" ihlulwe ngokwemiqathango yeemveliso ezithatha inxaxheba kuzo. Ngaloo ndlela kukho umyalelo wokuqala, wesibini, wesithathu okanye ngaphezulu. Zingakwazi ukwahlula kwiiklasi eziliqela: eziqhelekileyo kunye neziphumo ezikhethekileyo.

Kule phepha sibheka kuqala ukulandelelana okuqhelekileyo kokulinganisa okuqhelekileyo. Imizekelo kunye neendlela zokuzisombulula ziya kuxoxwa ngazo kumacandelo alandelayo. Siza kuqwalasela kuphela i-ODE, kuba le yimizekelo eqhelekileyo yokulingana. Eziqhelekileyo zihlulwe zibe yi-subspecies: ngokwahlukana kwezinto ezihlukeneyo, ezihambelanayo kunye nezixhepha. Emva koko, uya kufunda indlela ahluke ngayo, kwaye ufunde indlela yokuyicombulula ngayo.

Ukongezelela, ezi zilinganiso zinokudibaniswa ukuze emva kokuba sifumane inkqubo yokulinganisa okokuqala. Siza kujonga kwakhona iinkqubo kwaye sifunde indlela yokuzisombulula.

Kutheni sifundela kuphela umyalelo wokuqala? Ngenxa yokuba kufuneka uqale ngomntu olula, kwaye akunakwenzeka ukuchaza yonke into ehambelana nokulinganisa ngokwahlukileyo kwinqaku elilodwa.

Ukulingana kunye neenguqu ezihlukeneyo

Zizo, mhlawumbi, ukulinganisa okulula kokulandelana kokuqala. Ezi ziquka imizekelo engabhalwa njenge: y '= f (x) * f (y). Ukusombulula eli lingano sifuna ifomula yokumela i-derivative njengomlinganiselo wohlukeneyo: y '= dy / dx. Ngosizo lwaso sifumana ukulingana okulandelayo: dy / dx = f (x) * f (y). Ngoku siyakwazi ukuphendukela kwindlela yokuxazulula imimiselo eqhelekileyo: sihlula iinguqu ngeengxenye, oko kukuthi, sidlulisela yonke into ukusuka kwintlobo yintlobo yendawo apho i-dy ifumaneka khona, kwaye senza oku ngokuguquguquka x. Sifumana ukulingana kwefaysi dy / f (y) = f (x) dx, esetombululwa ngokuthatha inxaxheba kumacala omabini. Musa ukulibala ngokuqhubekayo, okumele kusekwe emva kokuthatha inxaxheba.

Isisombululo sawo nawuphi na "umsakazo" ngumsebenzi wokuxhomekeka kwe x xa (kwimeko yethu) okanye, ukuba kukho imeko yesibalo, impendulo ifom yefom. Makhe sihlalutye ngomzekelo onokhenkce wekhosi yonke yesisombululo:

Y '= 2y * isono (x)

Sithumela iinguqu kwiindlela ezahlukeneyo:

Dy / y = 2 * isono (x) dx

Ngoku sithatha i-integrals. Zonke ziyafumaneka kwitafile ekhethekileyo yokudibanisa. Kwaye sifumana:

Ln (y) = -2 * i-cos (x) + C

Ukuba kuyimfuneko, sinokubonisa "igruk" njengomsebenzi we "X". Ngoku singatsho ukuba ukulingana kwethu ngokuhlukileyo kuxazululwe ukuba loo mqathango ayicacisiwe. Umqathango unokunikezelwa, umzekelo, y (n / 2) = e. Emva koko sithatha indawo yezinto eziguqukileyo kwisisombululo kwaye sifumane ukubaluleka kokuqhubekayo. Ngokomzekelo wethu, ngu-1.

Ukulingana kokuqala kokungafani ngokulandelana

Ngoku uye kwiindawo eziyinkimbinkimbi. Ukulingana kokuqala kokulandelana kwamanqanaba angabonakaliyo kubhalwe kwifomu jikelele ngokulandelayo: y '= z (x, y). Kufuneka kuqatshelwe ukuba umsebenzi ochanekileyo weenguqu ezimbini uhambelana, kwaye awukwazi ukwahlula kwizinto ezixhomekeke kuzo: z ukusuka x kwaye z ukusuka y. Ukujonga ukuba i-equation ihambelana noko okanye ayikho, kuyinto elula: senza indawo yendawo x = k * x and y = k * y. Ngoku sinqumla konke k. Ukuba zonke ezo ncwadi ziyancitshiswa, ngoko-equation ihambelana kunye kwaye unokwenza isisombululo sayo ngokuphepha. Ukuhamba phambili, masithi: umgaqo wokusombulula le mizekelo ilula kakhulu.

Simele senze indawo endaweni: y = t (x) * x, apho umsebenzi oxhomekeke kwi x. Emva koko siyakwazi ukubonisa i-derivative: y '= t' (x) * x + t. Ukuhambisa konke oku kwi-equation yethu yasekuqaleni nokuyilula, sifumana umzekelo ngeendlela ezihlukeneyo kunye no-x. Siyicombulula kwaye sinokuxhomekeka kuye (x). Xa sifunde, yifake endaweni y = t (x) * x endaweni yethu yangaphambili. Emva koko sifumana ukuxhomekeka kwe y x.

Ukwenza kucace ngakumbi, makhe sithathe umzekelo: x * y '= yx * e ^{y / x} .

Ukukhangela ukutshintshwa konke kuncitshiswa. Ngenxa yoko, i-equation ihambelana ngokwenene. Ngoku senza enye indawo endaweni esithetha ngayo: y = t (x) * x kunye y '= t' (x) * x + t (x). Emva kokulula, sifumana ukulingana okulandelayo: t '(x) * x = -e ^t . Siyayisombulula umzekelo wokugqibela kunye nezixhobo ezihlukeneyo kwaye ufumane: e- ^t = ln (C * x). Kuphela kuphela ukuba sithathe indawo y y x x (ngokuba y = t * x, ke t = y / x), kwaye siyafumana impendulo: e ^{-y / x} = ln (x * C).

Ukulinganisa okuhlukileyo kokulandelana kokuqala

Ixesha lokuqwalasela esinye isihloko esibanzi. Siza kuhlalutya umyalelo wokuqala wokulinganisa okungafaniyo. Zihluke njani kwiimbini zangaphambili? Masibhale phantsi. Ukulinganisa okungafaniyo kohlu lokuqala kohlobo luyakubhalwa ngefomu jikelele ngokulingana okulandelayo: y '+ g (x) * y = z (x). Kubalulekile ukucacisa ukuba i (x) kunye ne-g (x) ingaba ubuninzi bexesha.

Kwaye ngoku umzekelo: y '- y * x = x ² .

Kukho iindlela ezimbini zokusombulula, kwaye siza kujongana nazo zombini. Iyokuqala yindlela yokutshintshana kwamaxesha angabonakaliyo.

Ukuze kulungiswe ukulingana ngale ndlela, kuyimfuneko yokuqala ukulinganisa i-side-side side to zero kwaye uphendule ukulingana okubangela ukuba emva kokudluliselwa kwamalungu kuthatha ifom:

Y '= y * x;

Dy / dx = y * x;

Dy / y = xdx;

Ln | y | = x 2/2 + C;

Y = e ^{x2 / 2} * kwi ^C = C ₁ * e ^{x2 / 2} .

Ngoku sifuna ukufaka indawo yeC1 rhoqo nge-v v (x), esiyifumanayo.

Y = v * e ^{x2 / 2} .

Sithatha indawo ye-derivative:

Y '= v' * e ^{x2 / 2-} x * v * e ^{x2 / 2} .

Uze ufakele la magama kwi-equation yasekuqaleni:

* ^{X2 / 2} - x * v * e ^{x2 / 2} + x * v * e ^{x2 / 2} = x ² .

Kuyabonakala ukuba kwicala lasekhohlo imigca emibini ikhansela. Ukuba ngomzekelo othile oku akuzange kwenzeke, ngoko wenze into engalunganga. Masiqhubeke:

'E ^{x2 / 2} = x2.

Ngoku siyazixazulula ukulingana okuqhelekileyo, apho sifuna ukwahlula iinguqu:

Dv / dx = x ² / e ^{x2 / 2} ;

Dv = x2 * e- ^{x2 / 2} dx.

Ukuze sikhiphe umbandela, kufuneka sisebenzise ukuhlanganiswa ngamalungu. Nangona kunjalo, oku akusiyo isihloko sesihloko sethu. Ukuba unomdla, unokufunda indlela yokwenza ngayo. Akunzima, kwaye nekhono elaneleyo kunye neengqalelo akuthathi ixesha elide.

Makhe siphendule kwindlela yesibini yokusombulula ukulingana okungahambelaniyo: indlela yeBernoulli. Yiyiphi indlela ekhawuleza kwaye kulula - ifike kuwe.

Ngoko, xa sisombulula i-equation ngale ndlela, kufuneka senze indawo endaweni: y = k * n. Lapha k kunye n nemisebenzi ethile kuxhomekeke kwi x. Emva koko i-derivative iya kubonakala ngathi: y '= k' * n + k * n '. Sithatha indawo yomibini kwisabelo:

K '* n + k * n' + x * k * n = x ² .

Iqela:

K '* n + k * (n' + x * n) = x ² .

Ngoku sifuna ukulinganisa nantoni ekhoyo kubakaki. Ngoku, ukuba sidibanisa izimbini ezibini ezibangelwayo, sifumana inkqubo yokulinganisa okokuqala, okufuneka ixazululwe:

N '+ x * n = 0;

K '* n = x ² .

I-equation yokuqala ixazululwa njenge-equation ejwayelekile. Ukwenza oku, kufuneka uhlukanise iziguquko:

Dn / dx = x * v;

Dn / n = xdx.

Sithatha inxaxheba kwaye sifumane: ln (n) = x 2/2. Emva koko, ukuba sichaza n:

I = e ^{x2 / 2} .

Ngoku sithatha indawo yokulingana okubangelwa kukulingana kwesibini kwenkqubo:

K '* e ^{x2 / 2} = x2.

Kwaye ukuguqula, sinokulingana okufanayo njengendlela yokuqala:

Dk = x ² / e ^{x2 / 2} .

Kwakhona asiyi kuphinda sichitha izenzo ezongezelelweyo. Kufanelekile ukuthetha ukuba okokuqala isisombululo sokulinganisa kokuqala ngokulandelana kwamabangela ubunzima obunzima. Nangona kunjalo, ngokucwiliswa ngokujulile kwinqaku, kuqala ukuphucula nokubhetele.

Ziziphi izilinganiso ezahlukileyo ezisetyenziswayo?

Ukulinganisa kakhulu okuhlukeneyo kusetyenziswa kwi-physics, ekubeni phantse yonke imithetho eyisiseko ibhaliwe kwifomu eyahlukileyo, kwaye ezo zifomula esizibonayo sisisombululo salolu linganiso. Kwi-chemistry, zisetyenziselwa isizathu esifanayo: imithetho eyisiseko ifunyenwe ngoncedo lwabo. Kwi-biology, ukulinganisa okuhlukeneyo kusetyenziswa ukubonisa indlela yokuziphatha kweenkqubo, umzekelo umxhasi-wenyama. Zingasetyenziselwa ukudala imodeli yokuvelisa, yithi, ikholoni yamachiza ezincinci.

Ulwahlulo oluthile luya kunceda njani ebomini?

Impendulo yalo mbuzo ilula: akukho ndlela. Ukuba awuyena usosayensi okanye unjiniyela, akunakwenzeka ukuba luncedo kuwe. Nangona kunjalo, ukuphuhliswa ngokubanzi, akunangqiqo ukuqonda ukuba yeyiphi i-equation equation and how is solved. Kwaye ke umbuzo wonyana okanye intombi "yintoni isahluko sokuhlukana?" Musa ukubeka kwi-cul-de-sac. Ewe, ukuba usosayensi okanye injiniya, wena ngokwakho uqonda ukubaluleka kwesi sihloko kwiphina isayensi. Kodwa into eyona nto kukuba ngoku umbuzo "indlela yokusombulula ukulinganisa ngokulinganayo kokuqala?" Unako ukuhlala uphendula. Vumelana, kuhlala kukuhle, xa uqonda oko abantu bayakoyika ukuyiqonda.

Iingxaki eziphambili kwisifundo

Ingxaki ebalulekileyo ekuqondeni kwesi sihloko isakhono esilungileyo sokudibanisa nokuhlukanisa imisebenzi. Ukuba awuyi kuthatha iziphumo kwaye uzibandakanya kakubi, mhlawumbi, kulungele ukufunda, ukuqonda iindlela ezahlukeneyo zokuhlanganiswa kunye nokwahlukana, kwaye ke ngoko kuphela ukuqala ukufunda iincwadi ezichazwe kwinqaku.

Abanye abantu bayamangalisa xa befunda ukuba i-dx ingadluliselwa, kuba ngaphambili (esikolweni) kwafunwa ukuba i-fraction / dx ayibonakali. Apha kufuneka ufunde iincwadi ezivela kwi-derivative kwaye uqonde ukuba umlinganiselo wexabiso elingenamlinganiselo ongasetyenziswa ekuxazululeni ukulingana.

Abantu abaninzi abaqapheli ngokukhawuleza ukuba ukuxazulula ukulinganisa kokuqala ukulandelana ngokuqhelekileyo kusebenze okanye kungabandakanyekanga, kwaye ukuxakisa kubanika inkathazo eninzi.

Yintoni enye ongayifunda ukuze uqondwe ngcono?

Kukulungele ukuqala ukucwiliswa okunye kwilizwe lokubala okungafaniyo kwiincwadi zezifundo ezizodwa, umzekelo, ekuhlalutheni lweemathematika kubafundi bezakhono ezingezizo zeemathematika. Emva koko unako ukuya kwincwadi ekhethekileyo.

Kufanelekile ukukhankanya ukuba, ngaphezu kokulinganisa okuhlukileyo, kukho ukulingana okulinganayo, ukuze uhlale unento yokuzama nokufundisisa.

Isiphelo

Siyathemba ukuba emva kokufunda eli nqaku, unengcamango yokuba yiyiphi imilinganiselo eyahlukileyo kunye nendlela yokuyicombulula ngokuchanekileyo.

Kukho nawuphi na, iimathematika nganoma iyiphi indlela ewusizo kuthi ebomini. Yenza ingqiqo kunye nengqwalasela, ngaphandle kwayo yonke into efana nayo ngaphandle kwezandla.