Ukuthuthukiswa kanye eyohlobo umehluko kwesibalo ye-oda kuqala. izibonelo izixazululo

Ngicabanga ukuthi kufanele kuqale umlando ezikhazimulayo ithuluzi zezibalo njengoba umehluko zibalo. Njengazo zonke umehluko kanye calculus ebalulekile, lezi zibalo yasungulwa ngo Newton ngasekupheleni kwekhulu le-17. Wayekholelwa kwaba ukutholakala kwakhe kubaluleke kangaka ukuthi ngisho umyalezo ibethelwe, okuyinto namuhla angahunyushwa ngalendlela lelandzelako: ". Zonke imithetho yemvelo esachazwa umehluko zibalo" Kungase kubonakale kuyihaba, kodwa kuyiqiniso. Noma yimuphi umthetho-physics, i-chemistry, i-biology, lingachazwa nge lezi zibalo.

Negalelo elikhulu ekuthuthukiseni nokudalwa inkolelo-mbono yokuziphendukela umehluko zibalo kufanele wezibalo of Euler futhi Lagrange. Kakade ekhulwini le-18 bathola futhi ithuthukiswe manje ukuthi ukutadisha ngesikhathi abaphezulu izifundo nyuvesi.

Isenzakalo esiyingqopha-mlando entsha ekuhloleni umehluko zibalo waqala sibonga Anri Puankare. He qamba "inkolelo-mbono qualitative ka umehluko zibalo", okuyinto, ngokuhlanganiswa imfundiso yokuziphendukela imisebenzi lokwehlukile eziyinkimbinkimbi nesandla kakhulu ukuze isisekelo yomphindwa - isayensi Isikhala impahla yayo.

Yiziphi umehluko zibalo?

Abantu abaningi besaba khona ibinzana "umehluko equation". Nokho, kulesi sihloko sizobe ebekwe ngokuningiliziwe ngokuyisisekelo lokhu ithuluzi eliwusizo kakhulu zezibalo okuyinto empeleni hhayi njengoba eziyinkimbinkimbi njengoba kubonakala kusukela isihloko. Ukuze uqale ukukhuluma ngento lokuqala oda umehluko equation, kuzomele uqale bajwayelane imiqondo eziyisisekelo abazalwa ezihlobene nale ncazelo. Futhi sizoqala kanye umehluko.

umehluko

Abantu abaningi bayazi leli gama kusukela esikoleni esiphakeme. Nokho, namanje bahlale ngokuningiliziwe. Cabanga igrafu umsebenzi. Asikwazi uwakhuphule kuze kube sezingeni lokuthi kukhona okunganqamula uthando ingxenye yayo iba umugqa oqondile. Kuyothatha amaphuzu amabili angaphezu ngokungenamkhawulo eduze komunye nomunye. Umehluko phakathi izixhumanisi zabo (x noma y) kuncane. Futhi kubizwa ngokuthi umehluko nezinhlamvu uqoke dy (umehluko y) kanye dx (umehluko x). Kubalulekile ukuqonda ukuthi umehluko akuyona ekugcineni ukubaluleka, futhi lena incazelo kanye umsebenzi oyinhloko.

Futhi manje kumele ucabangele izinto ezilandelayo, okuyinto thina kutawudzingeka sibachazele umqondo umehluko equation. It - esuselwe.

esuselwe

Sonke kumelwe ukuba wezwa esikoleni futhi lo mqondo. Bathi esuselwe - kuyinto nezinga lokukhula noma ukwehlisa umsebenzi. Nokho, le ncazelo kuba edidayo. Ake ngizame ukukuchazela imigomo esuselwe nomehluko. Ake ubuyele emuva kuncane umsebenzi isikhawu ngamaphuzu amabili, okuyinto zitholakala kude okungenani komunye nomunye. Kodwa ngisho nangale lo msebenzi ibanga kuyisikhathi sokushintshela kunenzuzo ethile. Futhi ukuchaza lolo shintsho futhi beveza esuselwe ebesingase sifakwe kuyi bunjesi isilinganiso nomehluko: f (x) '= DF / dx.

Manje-ke kubalulekile ukucabangela sezindawo eziyisisekelo esuselwe. Kukhona kuphela ezintathu:

1. isamba esuselwe noma umehluko kungenziwa imelelwa isamba noma umehluko we-derivatives: (a + b) '= a' + b ', futhi (ab)' = a'-b '.
2. Impahla yesibili ehlangene ukubuyabuyelela. senze imisebenzi esuselwe - yinani yemisebenzi omunye umsebenzi kwenye esuselwe: (a b *) '= a' * b + a * b.
3. I esuselwe umehluko singatlolwa bunjesi ezothando ezilandelayo: (a / b) '= (a' * ba * b ') / b ^2.

Zonke lezi zici singaba usizo kabi sokuthola izixazululo umehluko zibalo ye-oda kuqala.

Futhi, kukhona okunye esinqunyiwe. Ake sithi ube umsebenzi z, okuyinto incike eziguquguqukayo x kanye y. Ukubala esuselwe olungaphelele lo msebenzi, isibonelo, x, kudingeka sithathe y variable njalo futhi kulula ukuhlukanisa.

ebalulekile

Omunye umqondo obaluleke kakhulu - ebalulekile. Eqinisweni-ke kuphambene esuselwe. Integrals kukhona izinhlobo eziningana, kodwa izixazululo olula umehluko zibalo, sidinga eziwubala kakhulu integrals nanini.

Ngakho, liyini ebalulekile? Ake sithi sinalo ubuhlobo ezinye f x. Sithatha kuwo ebalulekile futhi ukuthola F umsebenzi (x) (ivamise ukubizwa ngokuthi bakudala), okuyinto esuselwe umsebenzi yokuqala. Ngakho-ke F (x) '= f (x). Lokhu kusho ukuthi i-ebalulekile esuselwe uyalingana umsebenzi original.

Ekuxazululeni umehluko zibalo kubaluleke kakhulu ukuba siqonde incazelo nemisebenzi ebalulekile, ngoba ngokuvamile kakhulu kufanele uzithatha ukuthola izixazululo.

I zibalo zihlukile kuye yimvelo yabo. Esigabeni esilandelayo kwethulwa sizobheka izinhlobo yokuqala oda umehluko zibalo, bese Ufunda indlela yokuxazulula kubo.

Kuqala amakilasi umehluko zibalo

"Diffury" ihlukaniswe ngomyalo nemikhiqizo kuzo. Ngakho kukhona oda wokuqala, wesibili, wesithathu nofana ngaphezulu. Lemisebenti ingaphindze yizigaba eziningana: abavamile futhi ngokwengxenye.

Kulesi sihloko, sizoxoxa abavamile umehluko zibalo ye-oda kuqala. Izibonelo kanye nezixazululo sixoxa zingxenye ezilandelayo. Sibheka kuphela TAC ngoba lwezinhlobo ezivame kakhulu of zibalo. Ordinary ihlukaniswe zenyathi: eziguquguqukayo separable, eyohlobo futhi heterogeneous. Okulandelayo uzofunda ihluke kanjani ngomunye nomunye, futhi ufunde indlela yokuxazulula kubo.

Ngaphezu kwalokho, lezi zibalo zingahlanganiswa, ukuze emva sithola isimiso umehluko zibalo ye-oda kuqala. izinhlelo ezinjalo, siphinde sibheke futhi ufunde indlela yokuxazulula.

Kungani izincwadi esizifundayo kuphela oda kuqala? Ngoba kubalulekile ukuqala elula ukuchaza zonke elihlobene ne umehluko zibalo, esihlokweni esisodwa akunakwenzeka.

Izibalo eziguquguqukayo separable

Lona mhlawumbe kakhulu elula kuqala ukuze umehluko zibalo. Lezi izibonelo ukuthi singatlolwa bunjesi: y '= f (x) * f (y). Ukuxazulula le kwesibalo sidinga ifomula umfanekiso esuselwe njengoba isilinganiso nomehluko: y '= dy / dx. Ngalo sithola ezothando: dy / dx = f (x) * f (y). Manje Singathandaza indlela yokuxazulula izibonelo ejwayelekile: ukwehlukanisa eziguquguqukayo ezingxenyeni, okusho phambili zonke y okuguquguqukayo engxenyeni lapho kukhona dy, futhi wenze x variable ... We ukuthola i equation yefomu: dy / f (y) = f (x) dx, okuyinto ibangelwa ngokuthatha integrals we izingxenye ezimbili. Ungakhohlwa mayelana njalo ukuthi ufuna ukubeka ngemuva ukuhlanganiswa.

Isixazululo iyiphi "diffura" - kuyinto umsebenzi x by y (kithi), noma uma kukhona isimo zezinombolo, impendulo inombolo. Ake sihlole isibonelo ukhonkolo yonke inkambo yokuhle isinqumo:

y '= 2y yesono (x)

Dlulisa eziguquguqukayo bacondze etindzaweni letehlukene:

dy / y = 2 yesono (x) dx

Manje thatha integrals. Bonke ingatholakala itafula olukhethekile integrals. Futhi sithola:

ln (y) = -2 * cos (x) + C

Uma kudingeka, singakwazi ukuveza "y" njengoba umsebenzi "X". Manje singasho ukuthi umehluko yethu kwesibalo isixazululekile, uma kungenjalo bucaciswe isimo. Kungenziwa isimo esibekiwe, isibonelo, y (n / 2) = e. Siyobe nje bamelele ukubaluleka kwalezi eziguquguqukayo in isinqumo futhi uthole ukubaluleka njalo. Ngo Isibonelo sethu, kungenxa 1.

Eyohlobo oda lokuqala umehluko zibalo

Manje on izingxenye eziyinkimbinkimbi. Eyohlobo kuqala ukuze umehluko zibalo ungalotshwa ifomu jikelele njengoba: y '= z (x, y). Kumele kuqashelwe ukuthi umsebenzi kwesokudla eziguquguqukayo ezimbili kuyinto iyunifomu, futhi ayikwazi lahlukaniswa kuye: z x kanye z ka-y. Hlola ukuthi ndaba kuyindlela eyohlobo noma cha, ulula: senza esikhundleni x = k * x kanye y = k * y. Manje thina usike wonke k. Uma lezi zincwadi aziwisa ke ezothando eyohlobo futhi kungaba ngokuphepha uqhubekele isixazululo yayo. Lapho ubheka ikusasa, sithi: isimiso yesixazululo salezi zibonelo futhi silula.

Sidinga ukwenza esikhundleni: y = t (x) * x, lapho t - umsebenzi futhi kuncike x. Khona-ke siyokwazi ukubonisa le esuselwe: y '= t' (x) * x + t. Uma kufakwa konke lokhu ku-equation sethu sokuqala bese lula ke, sinesibonelo lokwehlukaniswa lokwehlukile t njengoba x. Xazulula futhi ukuthola ukwencika t (x). Lapho sifika khona, umane bamelele esikhundleni sethu esidlule y = t (x) * x. Khona-ke sithola ukwencika y ku x.

Ukuze wenze kube sobala, siyowuqonda isibonelo: x * y '= yx * e y / x.

Lapho ehlola indawo yonke kuncipha. Ngakho, le ndaba kuyindlela eyohlobo ngempela. Manje wenze enye esikhundleni, sakhuluma ngokuthi: y = t (x) * x kanye y '= t' (x) * x + t (x). Ngemva yokuncishiswa ezothando okulandelayo: t '(x) * x = -e t. Sinquma uthole isampula eziguquguqukayo ehlukanisiwe futhi ^{sithola: e -t = ln (C *} x). Tsine lesifanele kukwenta nje esikhundleni t ngu y / x (ngoba uma y = t * x ke t = y / x), futhi sithola ^{impendulo: e -y / x = ln (} x * C).

Ukuthuthukiswa umehluko kwesibalo ye-oda kuqala

Sekuyisikhathi sicabangele esinye isihloko ezibanzi. Sizobheka heterogeneous lokuqala oda umehluko zibalo. Uyini umehluko kusukela odlule ezimbili? Asivume. Ukuthuthukiswa kuqala ukuze umehluko zibalo ifomu jikelele ezothando singatlolwa bunjesi ngaleyo ndlela: y '+ g (x) * y = z (x). Kumele sicacise ukuthi z (x) kanye g (x) kungenzeka amanani njalo.

Nasi isibonelo: y '- y * x = x ^2.

Kunezindlela ezimbili ukuxazulula, nathi esingabakhe oda Ake sihlole bobabili. Eyokuqala - indlela ukushintshashintsha constants ezingafanele.

Ukuze kuxazululwe le kwesibalo ngale ndlela, kubalulekile ukuba athi ngowokuqala ngakwesokudla kuqanda, nokuxazulula ezothando okuholela okuyinto emva ukudluliswa izingxenye iba:

y '= y * x;

dy / dx = y * x;

dy / y = xdx;

ln | y | = x ^2/2 + C;

y = e ^{X2 / 2} * ^C y = C ₁ * e ^{X2 / 2.}

Manje kubalulekile esikhundleni njalo C ₁ kwi v umsebenzi (x), okuyinto sizothola.

y = v * e ^{X2 / 2.}

Dweba esuselwe esikhundleni:

^{y '= v' * e X2} / ² -X * v * e ^{X2 / 2.}

Futhi esikhundleni lezi zinkulumo phakathi kwesibalo yasekuqaleni:

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

Ungabona ukuthi ohlangothini lwesobunxele imigomo emibili ziyancipha. Uma abanye isibonelo akwenzekanga lokho, khona-ke kukhona okubi esikwenzile. Siyaqhubeka:

v '* e ^{X2 / 2} = x ^2.

Manje thina ukuxazulula kwesibalo evamile lapho ufuna ukuhlukanisa eziguquguqukayo:

DV / dx = x ^{2 /} e ^X2 / ^2;

DV = x ² * e ^{- X2 / 2} dx.

Ukuze ususe ebalulekile, kufanele afake isicelo ukuhlanganiswa by izingxenye lapha. Nokho, lokhu akukhona isihloko lesi sihloko. Uma unentshisekelo, ungafunda bebodwa ukuba benze izimanga ezinjalo. Akunzima, futhi amakhono ngokwanele nokunakekela akuyona isikhathi.

Ebhekisela indlela yesibili nesixazululo zibalo inhomogeneous: indlela Bernoulli. Yini indlela kusheshe futhi kube lula - kuphuma kuwe.

Ngakho, nawurarulula le ndlela, kudingeka senze esikhundleni: y = k * n. Lapha, k futhi n - abanye imisebenzi kuye x. Khona-ke esuselwe izobukeka: y '= k' * n + k * n '. Obambele ezimbili izisusi kule ndaba:

k '* n + k * n ' + x * k * n = x ^2.

Iqembu i:

k '* n + k * ( n' + x * n) = x ^2.

Manje kubalulekile kufanisa kuqanda, okungukuthi kubakaki. Manje, uma sihlanganise lo zibalo ezimbili okuholela, sithola uhlelo oda lokuqala umehluko zibalo ukuba ixazululwe:

n '+ x * n = 0;

k '* n = x ^2.

I ukulingana lokuqala unqume ukuthi ezothando evamile. Ukuze wenze lokhu, kudingeka uhlukanise eziguquguqukayo:

dn / dx = x * v;

dn / n = xdx.

Sithatha ebalulekile futhi sithola: ln (n) = x ^2/2. Khona-ke, uma sibonisa n:

n = e ^{X2 / 2.}

Manje shintsha kwesibalo okuholela phakathi kwesibalo yesibili:

k '* e ^{X2 / 2} = x ^2.

Nokuguqula, sithola ezothando kuyafana indlela yokuqala:

dk = x ^{2 /} e ^X2 / ^2.

Siphinde ngeke uxoxe esinye isinyathelo. Kuthiwa ekuqaleni lokuqala oda umehluko zibalo isixazululo kubangela ubunzima obukhulu. Nokho, i-ukucwiliswa ngokujulile ngesihloko iyaqala ukuthola kangcono.

Ukuphi umehluko zibalo?

Kakhulu asebenzayo umehluko zibalo asetshenziswe physics, njengoba cishe yonke imithetho eyisisekelo zibhaliwe ifomu umehluko, nalabo amafomula, ukuthi sibona - ikhambi lezi zibalo. Ngo-chemistry, bajwayele ngenxa yesizathu esifanayo: imithetho eyisisekelo zihlobene ngabo. Ngo biology, i-zibalo umehluko zisetshenziselwa kukhombisa ukuziphatha izinhlelo, ezifana umhlaseli - inyamazane. Lemisebenti ingaphindze isetshenziselwe ukudala onobuhle yokuzala, isibonelo, amakoloni nezilwanyana ezincane.

Njengoba umehluko zibalo ukusiza ekuphileni?

Impendulo yalo mbuzo ilula: lutho. Uma ungeyena usosayensi noma unjiniyela, akunakwenzeka ukuthi kuyoba usizo. Nokho, hhayi ubuhlungu ukwazi ukuthi ezothando umehluko futhi isixazululekile sokuthuthukiswa jikelele. Bese umbuzo indodana noma indodakazi, "lokho ezothando umehluko?" ungafaki ngawe ekupheleni efile. Hhayi-ke, uma usosayensi noma unjiniyela, lapho uyazi ukubaluleka lesi sihloko iyiphi isayensi. Kodwa okubaluleke kakhulu, manje lo mbuzo, "indlela yokuxazulula ezothando umehluko oda kuqala?" uyokwazi ukunikeza impendulo ngaso sonke isikhathi. Ngiyavuma, kuba njalo nice uma ubona ukuthi lokho okucatshangwa abantu ngisho esabe ukuthola.

Izinkinga main kulolu cwaningo

Inkinga eyinhloko ekuqondeni isihloko umkhuba othile omubi kakhulu ekuhlanganiseni naletinye kwamangqamuzana imisebenzi. Uma ungakhululekile lokucabanga nemikhiqizo kanye integrals, kuba cishe nibaluleke ukufunda, ukufunda izindlela ezahlukene ukuhlanganiswa kanye kwamangqamuzana, futhi kuphela bese uqhubekela phambili ekutadisheni indaba iye yachazwa ezinhlwini ezingu-athikili.

Abanye abantu bayamangala uma bethola ukuthi dx angadluliselwa, njengoba ngaphambilini (esikoleni) wagomela ukuthi ingxenyenamba dy / dx kuyinto indivisible. Khona-ke kudingeka ufunde izincwadi esikhumulweni esuselwe futhi baqonde ukuthi isimo sengqondo ngobuningi ngokungenamkhawulo ezincane, okungase abasetshenziswa ekuxazululeni zibalo.

Abantu abaningi abazi ngokushesha bayaqaphela ukuthi ikhambi of umehluko zibalo ye-oda kuqala - lokhu ngokuvamile umsebenzi noma neberuschiysya ebalulekile, futhi kulokhu ukuduka ibanika asemrarweni.

Yini enye zihlolwe ukuze uqonde kangcono?

Kungcono ukuqala ukucwiliswa okwengeziwe ezweni umehluko calculus izincwadi ezikhethekile, isibonelo, umqulusizinda abafundi amakhono angewona awe-zezibalo. Ungabese ukuhambisa lezi zincwadi ezikhethekile ngaphezulu.

Kuthiwa, ngaphezu umehluko, kusenabantu zibalo ebalulekile, ngakho Kuyohlale kunento okufanele siyiphishekele nokufanele ukuthi ukufunda.

isiphetho

Sithemba ukuthi ngemva kokufunda lesi sihloko ngeke une umqondo walokho zibalo umehluko futhi indlela yokuxazulula kubo ngendlela efanele.

Kunoma ikuphi, izibalo nganoma iyiphi indlela ewusizo ekuphileni kwethu. Kutfutfukisa logic nokunakwa, ngaphandle okuyinto wonke umuntu, njengoba elingenziwe ngezandla. '