Linear uye homogeneous differential mukana wokuti vanhu vanga wokutanga kurongeka. mienzaniso zvingaitwa

Ndinofunga tinofanira kutanga nhoroondo mbiri kwemasvomhu chishandiso sezvo differential equations. Kufanana ose differential uye chinokosha calculus, equations aya afunga Newton mukupera kwezana remakore rechi17. Aidavira yakanga ake Kuwanika zvichikosha kuti kunyange encrypted shoko, iro nhasi anogona kushandurwa sezvinotevera: ". All mitemo yezvinhu zvakasikwa zvinorondedzerwa differential equations" Zvingaita sokuti kuwedzeredza, asi zviri zvechokwadi. Chero murairo wefizikisi, kemisitiri, zvipenyu, inogona akarondedzerwa equations izvi.

An mupiro kukuru kuti kukura uye pakusika dzidziso differential equations vane samhu Euler uye Lagrange. Nechekare muzana remakore rechi18 vakaona uye yakatanga chii zvino kudzidza panguva mukuru yunivhesiti mapoka.

A mberi itsva kudzidza differential equations akatanga kuonga Anri Puankare. Akasika munhu "Qualitative dzidziso differential equations", iyo, pamwe chete dzidziso mashandiro anoita variables akaoma akabatsira yekudyidzana kusvikira nheyo topology - sayenzi nzvimbo yayo zvivako.

Ndezvipi differential equations?

Vanhu vazhinji vanotya okuti "differential mukana wokuti vanhu vanga". Zvisinei, munyaya ino isu Ndichasimuka zvakadzama musimboti ichi unobatsira chaizvo kwemasvomhu chokushandisa chiri chaizvoizvo asina kuoma sezvo kunenge kubva rokuremekedza. Kuti kutaura nezvenyaya vomuzana kuitira differential mukana wokuti vanhu vanga, unofanira kutanga tinyatsoziva dzinokosha pfungwa akangosikwa nechokuita nerondedzero iyi. Uye isu tichati kutange differential.

differential

Vanhu vakawanda vanoziva izwi ichi kubva chesekondari. Zvisinei, achiri kugara pamusoro payo zvizere. Fungidzira girafu pakati mashandiro. Tinogona kuwedzera kuti zvokuti chero zvayo chidimbu inova achirurama mutsetse. It mukatora mbiri pfungwa kuti vari zvikuru pedyo nepedyo. Musiyano Coordonatele dzavo (x kana Y) ndiyo kamwoyo. Uye inonzi differential uye vatambi kunan'anura Dy (differential kuti ja) uye yori (the differential kuti x). Zvakakosha kunzwisisa kuti differential haasi mukuru kukosha, uye ichi ndicho zvinoreva uye chikuru basa.

Zvino iwe unofanira inotevera okunze, izvo tichada kutsanangura differential mukana wokuti vanhu vanga pfungwa. It - rinobva.

rinobva

Tose tinofanira kunge akanzwa kuchikoro uye ichi pfungwa. Vanoti rinobva - ndiwo mwero kukura kana kuderera kuti mashandiro. Zvisinei, tsanangudzo iyi anova kuvhiringidza. Ngatiedzei kutsanangura rinobva Pakuurayiwa differentials. Ngatidzokerei kune kamwoyo penguva mashandiro maviri pfungwa, izvo riri pana risingamboshandisi kure nomumwe. Asi kunyange kupfuura ichi kure basa ndiyo nguva change vamwe kukosha. Uye kutsanangura kuti kuchinja uye kukwira rinobva izvozvo zvimwe kunyorwa sezvo nhamba yacho differentials: f (x) '= df / yori.

Zvino zvakafanira kuti funga dzinokosha ehupfumi ari rinobva. Kune zvitatu chete:

1. Rinobva Verengai kana musiyano inogona achimiririrwa Verengai kana musiyano pakati nomukaka: (a + b) '= munhu' + B ', uye (AB)' = a'-b '.
2. The pfuma chechipiri hwakabatana chokuwanziridza. Rinobva mabasa - ndiko kuwanda mabasa rimwe basa kune imwe rinobva: (a * b) '= munhu' b + mumwe * B '.
3. The rinobva musiyano anogona kunyorwa seinotevera mukana wokuti vanhu vanga: (a / b) '= (a' * ft * B ') / b ^2.

All idzi kuuya batsira kuwana mhinduro differential equations wokutanga kurongeka.

Uyewo, pane kukwana nomukaka. Ngatitii tine kushanda Z, iyo hunotsigirwa variables X uye Y. Kuverenga haasaruri rinobva pane basa iri, somuenzaniso, mu X, tinofanira kutora shanduka and kuramba uye nyore kusiyanisa.

chakakosha

Chimwe pfungwa inokosha - chinokosha. Kutaura zvazviri ndicho kunopesana rinobva. Integrals mhando dzakawanda, asi nyore kugadzirisa zviri differential equations, tinoda zvikuru sezvisingakoshi nokusingagumi integrals.

Saka, chii chinokosha? Toti tine zvimwe ukama add kuti x. Isu kubva mairi chinokosha uye kuwana basa F (x) (chinowanzoonekwa kunzi chinyakare), iro rinobva pakutanga basa. Naizvozvo F (x) '= F (x). Izvi zvinoreva kuti chinokosha pamusoro rinobva akaenzana yepakutanga basa.

In nekugadzirisa differential equations zvakakosha chaizvo kuti tinzwisise uye kushanda chinokosha, sezvo kazhinji kacho kuti avaendese kuwana mhinduro.

The equations zvakasiyana zvichienderana hunhu hwavo. In chinotevera tichakurukura mhando pakutanga kurongeka differential equations, uye ipapo kudzidza kuagadzirisa.

Makirasi differential equations

"Diffury" yakakamurwa sezvakarayirwa nomukaka kuriitawo. Saka pane wokutanga, wechipiri, wechitatu kana kupfuura kuti. Vanogonawo kukamurwa kuva makirasi akawanda: misi uye kukwana.

Munyaya ino, tichakurukura vanhuwo differential equations wokutanga kurongeka. Mienzaniso uye negadziriso isu munyaya inotevera zvikamu. We funga voga TAC nokuti emhando dzakawanda zvinowanzovapo equations. Munhuwo rakakamurwa subspecies: pamwe kupatsanurwa variables, homogeneous uye heterogeneous. Next uchadzidza sei akasiyana mumwe, uye kudzidza sei kuagadzirisa.

Uyezve, equations izvi zvinogona zvabatanidzwa saka kuti pashure isu tiwane kwegadziriro differential equations wokutanga kurongeka. Kwakadaro hurongwa, isuwo kutarisa uye kudzidza kugadzirisa.

Nei tiri kufunga kutanga chete murayiro? Nokuti zvakafanira kuti kutanga nyore uye tsanangura zvose chokuita differential equations, mune rimwe munyaya hazvibviri.

Equations vane kupatsanurwa variables

zvichida ichi ndicho nyore yokutanga kuti differential equations. Ava mienzaniso kuti anogona kunyorwa se: y '= d (x) * F (Y). Kugadzirisa mukana wokuti vanhu vanga ichi tinoda mufananidzo payakavakirwa ari rinobva sezvo nhamba yacho differentials: y '= Dy / yori. Narwo tinowana mukana wokuti vanhu vanga: Dy / yori = d (x) * F (Y). Zvino tinogona kuenda nzira kugadzirisa vomureza mienzaniso: kuparadzanisa variables muzvikamu, kureva kutsanya mberi zvose shanduka and ari chikamu pane Dy, uye zvakare kuti shanduka X ... We kuwana mukana wokuti vanhu vanga vari fomu: Dy / F (Y) = F (x) yori, izvo zvinoitwa nokutora integrals wemigove maviri. Usakanganwa pamusoro dzose kuti uchida kuisa pashure kusangana.

The Kugadzirisa upi "diffura" - iri kushanda X na Y (kwatiri), kana pane nhamba ezvinhu, mhinduro ndiyo nhamba. Ngationgororei chesamende muenzaniso rose nzira chisarudzo:

Y '= 2y * chivi (x)

Uchatamisa variables kwakasiyana-siyana:

Dy / and = 2 * chivi (x) yori

Zvino tora integrals. Vose vanogona kuwanikwa rinokosha tafura integrals. Uye isu kuwana:

Dr (Y) = -2 * Kosi (x) + C

Kana waida, tinogona kutaura "and" ave kushanda "X". Zvino tinogona kutaura kuti kwedu differential mukana wokuti vanhu vanga riri kugadziriswa, kana kusingajekeswi ezvinhu. Inogona wainyatsotaura ezvinhu Somuenzaniso, y (n / 2) = e. Zvadaro tichava nyore kuisa ukoshi variables izvi zviri chisarudzo uye kuwana kukosha dzose. In muenzaniso wedu, ndizvo 1.

Homogeneous rokutanga kuti differential equations

Zvino kune mamwe kunzwisisa nhengo. Homogeneous rokutanga kuti differential equations anogona kunyorwa vakawanda muchimiro sezvo: y '= z (x, y). Zvinofanira kucherechedzwa kuti zvakarurama mashandiro variables vaviri yunifomu, uye haagoni huviri zvichienderana: z X uye z kuti ja. Ongorora kuti mukana wokuti vanhu vanga iri homogeneous kana kwete, ari nyore, isu kuti kuzvitsinhanisa x =, k * X uye ja =, k * and. Zvino isu akacheka K zvose. Kana tsamba idzi anodonhedzwa, ipapo mukana wokuti vanhu vanga homogeneous uye anogona zvakachengeteka mberi kwayo mhinduro. Achitarira mberi, tinoti: nheyo yokupedza mienzaniso iyi kuri nyore chaizvo.

Tinofanira kuita kuzvitsinhanisa: y = t (x) * X, apo T - munhu basa kuti inobvawo pane x. Zvadaro tinogona kutaura rinobva: y '= t' (x) * X + t. Dzinomiririra izvi zvose munzvimbo mukana wokuti vanhu vanga vedu vokutanga uye kurerutsa nayo, tine muenzaniso kuparadzana variables T x. Kurigadzirisa uye kuwana kutsamira T (x). Pataisvika izvozvo, inongova kutsiva dzedu dzekare kuzvitsinhanisa Y = t (x) * x. Zvadaro isu tiwane kuvimba kuti ja pamusoro x.

Kuti izvozvo zvakajeka, patichanzwisisa muenzaniso: x * Y '= yx * E and / x.

Kana kuongorora yaitsiva zvose kuderera. Saka, mukana wokuti vanhu vanga zviri homogeneous. Zvino itai chimwe kuzvitsinhanisa, takataura pamusoro: y = t (x) * X uye Y '= t' (x) * X + T (x). Mushure anoitwa inotevera mukana wokuti vanhu vanga: t '(x) * X = -e t. Tinosarudza kuwana mharidzo ine kumwedzi variables uye isu ^{tiwane: e -t = Dr (C *} x). Tinofanira chete tinoda kutsiva T kubudikidza ja / X (nekuti kana ja = T * X, ipapo T = y / x), uye isu tiwane ^{mhinduro: e -y / x = Dr (} X * C).

Linear differential mukana wokuti vanhu vanga wokutanga kurongeka

Ndiyo nguva yokuti funga mumwe yakafara musoro. Tichatarisa heterogeneous vomuzana kuitira differential equations. Sei vakasiyana yapfuura maviri? Ngatimbotaurei zvazviri. Linear rokutanga kuti differential equations ari mukuru muchimiro mukana wokuti vanhu vanga anogona kunyorwa seizvi: y '+ d (x) * Y = z (x). Zvinofanira yakajekeswa kuti z (x) uye d (x) kungava kuramba tsika.

Heuno muenzaniso: y '- Y * x = X ^2.

Pane nzira mbiri kugadzirisa, uye zvatinorayira Ngationgororei vaviri. Chokutanga - the nzira kusiyana zvemasanga constants.

Kugadzirisa mukana wokuti vanhu vanga saizvozvi, zvakakodzera inofananidza wokutanga kurudyi kuna zero, uye kugadzirisa matambudziko mukana wokuti vanhu vanga iyo pashure kuchinjwa zvikamu zvava:

Y '= y * X;

Dy / yori = y * X;

Dy / Y = xdx;

Dr | y | = X ^2/2 + C;

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

Zvino zvakafanira kuti kutsiva dzose C ₁ pamusoro mashandiro V (x), izvo isu muchawana.

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

Swedera yokutsiva rinobva:

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

Uye dzinomiririra mashoko aya munzvimbo yepakutanga mukana wokuti vanhu vanga:

V '* e ^{x2 / 2} - X * vs. * e ^{x2 / 2} + X * vs. * e ^{x2 / 2} = X ^2.

Unogona kuona kuti ari kurutivi mazwi maviri kuruboshwe vari ipfupi. Kana vamwe muenzaniso kuti hazvina kuitika, zvino waita chinhu chakaipa. Tikaramba:

V '* e ^{x2 / 2} = X ^2.

Zvino isu vagadzirise zvatinosiita mukana wokuti vanhu vanga yauri kuda kuparadzanisa variables:

DV / yori = X ^{2 /} ja ^x2 / ^2;

DV = X ² * e ^{- x2 / 2} yori.

Kubvisa chinokosha, tinofanira kushandisa kusangana ne zvikamu pano. Zvisinei, izvi hazvina nyaya ino. Kana uchida, unogona kudzidza pachavo kuita zviito zvakadaro. Hazvina kuoma, uye nounyanzvi zvakakwana uye nokutarisira haisi nguva unoparadza.

Achitaura nzira wechipiri mhinduro yacho inhomogeneous equations: Bernoulli nzira. Inzira ipi nokukurumidza uye nyore - zviri kwamuri.

Saka, kana kugadzirisa nzira iyi, tinofanira kuita kuzvitsinhanisa: y = K * n. Pano, K N - mamwe mabasa zvichienderana x. Ipapo rinobva vachatarira akafanana: y '= K' * N + K * n '. Okutsiva vokutsiva vaviri mukana wokuti vanhu vanga:

K '* N + K * N ' + X *, k * n = X ^2.

Group kumusoro:

K '* N + K * ( n' + X * n) = X ^2.

Zvino zvakafanira kuti inofananidza kuna zero, zviri muzvikomberedzo. Zvino, kana imi nokubatanidza maviri zvichiguma equations, tinowana kwegadziriro rokutanga kuitira differential equations kuti kugadziriswa:

n '+ X * n = 0;

K '* n = X ^2.

The kuenzana wokutanga vanosarudza sei inoda mukana wokuti vanhu vanga. Kuti aite izvi, unofanira kuparadzanisa variables:

dn / yori = X * ndima;

dn / n = xdx.

Tinotora chinokosha uye tinowana: Dr (n) = X ^2/2. Zvadaro, kana isu ataure N:

n = E ^{x2 / 2.}

Zvino kutsiva kunoguma mukana wokuti vanhu vanga munzvimbo yechipiri mukana wokuti vanhu vanga:

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

Uye kuchinja, tinowana chete mukana wokuti vanhu vanga sezvinoitwa nzira yokutanga:

DK = X ^{2 /} ja ^x2 / ^2.

Isu handizonzwiwo kukurukura mberi nechiito. Zvinonzi panguva wokutanga-kuitira differential equations mhinduro kunokonzera matambudziko yakawanda. Zvisinei, zvakadzama mumvura iri nyaya iri kutanga kuti zviri nani.

differential equations varipiko?

Chaizvo differential equations kushandiswa wefizikisi, sezvo anenge zvose mitemo inokosha akanyorwa differential chimiro, uye Masvomhu avo, kuti tinoona - kugadzirisa equations izvi. In wekemisitiri, vari kushandiswa chete chikonzero: mitemo dzinokosha hwokurara kuburikidza navo. In zvipenyu, zviri differential equations zvinoshandiswa zvinoedza maitiro hurongwa, akadai chikara - zvakatapwa. Vanogonawo kushandiswa pave yokutevedzera wokuberekana Somuenzaniso, mapoka tusingaoneki.

Sezvo differential equations kubatsira muupenyu?

Mhinduro yomubvunzo uyu nyore: hapana. Kana usiri musayendisiti kana injiniya, hazvingaiti kuti vachava chinobatsira. Zvisinei, hachingatongovakuvadzi kuziva zvinotaurwa differential mukana wokuti vanhu vanga uye kugadziriswa kuti zvinonyatsorehwa kukura. Uye mubvunzo mumwe mwanakomana kana mwanasikana, ipapo "kuti differential mukana wokuti vanhu vanga?" regai kuisa iwe mune akafa magumo. Zvakanaka, kana uri musayendisiti kana injiniya, unozoziva kukosha wenyaya iyi chero sainzi. Asi chinonyanya kukosha, kuti zvino mubvunzo "kugadzirisa differential mukana wokuti vanhu vanga rokutanga kurongeka?" iwe uchagara kupa mhinduro. Bvumiranai, nguva dzose zvakanaka paunoona kuti chii vanhu vari kunyange kutya kuziva.

Matambudziko makuru ari kudzidza

Dambudziko huru iri kunzwisisa nyaya iri tsika yakaipa kusangana uye Kusiyanisa mabasa. Kana iwe uri kusununguka vanoti nomukaka uye integrals, zviri zvimwe munokosha kudzidza, kudzidza nzira dzakasiyana siyana kusangana uye Kusiyanisa, uye chete ipapo mberi kune kudzidza zvinhu kwave inorondedzerwa munyaya.

Vamwe vanhu vanoshamisika kudzidza kuti yori anogona kuchinjwa, sepakutanga (kuchikoro) vakati chikamu chiduku Dy / yori iri indivisible. Zvino unofanira kuverenga mabhuku pamusoro rinobva uye kunzwisisa kuti mafungiro zhinji zvikuru duku, izvo zvinogona kushandiswa kugadzirisa equations.

Vanhu vakawanda havana pakarepo kuti agadzirise zviri differential equations wokutanga kurongeka - ichi kazhinji kushanda kana neberuschiysya chinokosha, uye kurasika ichi anovapa taisireva.

Chiizve zvinogona kudzidzwa kunzwisisa zviri nani?

Zvakanakisisa kutanga zvimwe kunyudza munyika yose differential calculus kuti unyanzvi mabhuku Somuenzaniso, mune kwemasvomhu Kuongororwa kuti vadzidzi vasiri kwemasvomhu unyanzvi. Unogona kutamira zvikuru unyanzvi mabhuku.

Zvinonzi, kuwedzera differential, kuchine chinokosha equations, saka iwe nguva dzose uine chinhu kuedza kuti uye chii kudzidza.

mhedziso

Tinovimba kuti pashure kuverenga nyaya ino uve pfungwa zvinotaurwa differential equations uye sei kuagadzirisa nomazvo.

Chero zvazvingava, masvomhu chero nzira inobatsira kwatiri muupenyu. Inokudziridza kufunga uye kutarisirwa, pasina iro mumwe nomumwe, sezvo pasina namaoko.