Tawm thiab homogeneous differential kab zauv ntawm thawj thiaj li. piv txwv ntawm cov kev daws

Kuv xav tias peb yuav tsum pib nrog cov keeb kwm ntawm lub glorious zauv cuab tam li differential sib npaug. Zoo li tag nrho cov differential thiab integral calculus, cov equations tau invented los ntawm Newton nyob rau hauv lub lig 17th caug xyoo. Nws ntseeg tias nws yog nws foundations thiaj li tseem ceeb hais tias txawm lub encrypted lus, uas niaj hnub no muaj peev xwm muab txhais raws li nram no: "Tag nrho cov cai ntawm qhov piav los ntawm differential sib npaug." Nws kuj zoo li ib tug exaggeration, tab sis nws yeej muaj tseeb. Tej kev cai lij choj ntawm physics, Science News for KIDS, biology, yuav tau piav los no sib npaug.

Ib tug enormous pab mus rau txoj kev loj hlob thiab creation ntawm lub hom phiaj ntawm differential sib npaug muaj kev kawm txog zauv ntawm Euler thiab Lagrange. Twb tau nyob rau lub 18th xyoo pua lawv sab thiab tsim dab tsi yog tam sim no kawm nyob rau senior university kawm.

Ib tug tshiab milestone nyob rau hauv txoj kev tshawb no ntawm differential sib npaug pib ua tsaug rau Anri Puankare. Nws tsim ib tug "zoo xav ntawm differential sib npaug", uas, ua ke nrog lub hom phiaj ntawm kev khiav dej num ntawm txoj variables pab ho mus rau lub hauv paus ntawm topology - qhov kev kawm ntawm qhov chaw thiab nws cov khoom.

Yuav ua li cas yog differential sib npaug?

Muaj coob tus neeg ntshai ntawm cov lus "differential kab zauv". Txawm li cas los, nyob rau hauv no tsab xov xwm peb yuav teem tawm nyob rau hauv kom meej lub essence ntawm no heev pab tau xyuam xim lub cuab tam uas yog ua tau tsis yog raws li nyuab li no mas los ntawm lub title. Nyob rau hauv thiaj li yuav pib tham txog ib tug thawj-thiaj differential kab zauv, koj yuav tsum xub mus ntsib cov lub tswv yim uas yog inherently txuam nrog rau qhov no txhais. Thiab peb mam li pib nrog lub differential.

differential

Muaj coob tus neeg paub qhov no lub sij hawm txij li thaum high school. Txawm li cas los, tseem nyob rau ntawm nws nyob rau hauv kom meej. Xav txog tej yam lub teeb ntawm cov nuj nqi. Peb yuav ua rau kom nws mus cuag ib tug twg uas muaj ib yam ntawm nws ya ua ib tug ncaj kab. Nws mam li muab ob lub ntsiab lus hais tias yog infinitely ze rau txhua lwm yam. Qhov txawv nruab nrab ntawm lawv coordinates (x los yog y) yog infinitesimal. Thiab nws yog hu ua differential thiab cov cim xaiv dy (differential ntawm y) thiab dx (lub differential ntawm x). Nws yog ib qho tseem ceeb kom nkag siab txog hais tias cov differential tsis yog lub qhov kawg tus nqi, thiab qhov no yog lub ntsiab lus thiab lub ntsiab muaj nuj nqi.

Thiab tam sim no koj yuav tsum xav txog cov nram qab no hais, uas peb yuav tsum tau piav qhia txog lub differential kab zauv tswvyim. Nws - derivative.

derivative

Tag nrho cov ntawm peb yuav tsum tau hnov nyob rau hauv tsev kawm ntawv thiab qhov no notion. Lawv hais tias lub derivative - yog tus nqi ntawm txoj kev loj hlob los yog txo ntawm cov nuj nqi. Txawm li cas los, qhov no txhais yuav ntau ruam tag. Cia peb sim los piav cov derivative cov nqe lus ntawm lub differentials. Wb rov qab mus rau lub infinitesimal luv nuj nqi nrog ob lub ntsiab lus, uas yog nyob rau ntawm ib tug tsawg kawg nkaus deb ntawm txhua lwm yam. Tab sis txawm tshaj no deb muaj nuj nqi yog lub sij hawm mus hloov rau ib co nqi. Thiab los mus piav qhia hais tias kev hloov thiab tuaj mus nrog ib tug derivative uas yuav txwv tsis pub muab sau raws li tus piv ntawm cov differentials: f (x) '= df / dx.

Tam sim no nws yog tsim nyog los xav txog qhov yooj yim zog ntawm cov derivative. Muaj tsuas yog peb:

1. Derivative sum los yog qhov txawv yuav muaj tuaj raws li cov sum los yog sib txawv ntawm cov derivatives: (a + b) '= ib tug' + b ', thiab (b)' = a'-b '.
2. Qhov thib ob cuab yeej yog kev cob cog rua nrog npaug. Derivative tej hauj lwm - yog sum ntawm tej hauj lwm ntawm ib tug muaj nuj nqi rau lwm derivative: (a * b) '= ib tug' * b + ib * b '.
3. Lub derivative ntawm qhov txawv yuav sau raws li cov nram qab no kab zauv: (a / b) '= (ib tug' * kav ywm * b ') / b ^2.

Tag nrho cov nta tuaj nyob rau hauv muab rau nrhiav kev daws rau txawv sib npaug ntawm cov thawj kev txiav txim.

Tsis tas li ntawd, muaj ib nrab derivatives. Piv txwv peb muaj ib tug muaj nuj nqi ntawm lub z, uas yog nyob rau ntawm lub zog x thiab y. Los mus laij lub ib nrab derivative ntawm no muaj nuj nqi, piv txwv li, nyob rau hauv x, peb yuav tsum coj lub nce mus nce los y rau qhov thiab yooj yim qha.

ib

Ib qho tseem ceeb tswvyim - ib. Nyob rau hauv qhov tseeb nws yog qhov fab ntxeev ntawm derivative. Integrals ob peb hom, tab sis qhov nyuaj dlaws ntawm differential sib npaug, peb xav tau tshaj plaws tsis tseem ceeb indefinite integrals.

Yog li ntawd, dab tsi yog qhov tseem ceeb heev? Wb hais tias peb muaj ib co kev sib raug zoo f ntawm x. Peb coj los ntawm nws lub ib thiab muab tau ib muaj nuj nqi F (x) (nws yog feem ntau xa mus rau raws li ib tug txheej thaum ub), uas yog ib tug derivative ntawm tus thawj muaj nuj nqi. Yog li ntawd F (x) '= f (x). Qhov no kuj implies hais tias cov ib ntawm lub derivative yog sib npaug zos rau cov thawj muaj nuj nqi.

Nyob rau hauv kev daws differential sib npaug nws yog ib qho tseem ceeb heev kom to taub lub ntsiab lus thiab muaj nuj nqi ntawm lub integral, txij li thaum ntau heev tau coj lawv mus nrhiav kev daws teeb meem.

Lub sib npaug yog cov sib txawv nyob ntawm seb lawv qhov. Nyob rau hauv lub tom ntej no tshooj lus no peb yuav saib hom thawj thiaj li differential equations, thiab ces kawm ua li cas los daws kom tau lawv.

Cov chav kawm ntawm differential sib npaug

"Diffury" muab faib los ntawm qhov kev txiav txim ntawm derivatives muab kev koom tes nyob rau hauv lawv. Yog li muaj yog ib tug thawj, thib ob, thib peb los yog ntau tshaj kev txiav txim. Lawv muaj peev xwm kuj yuav muab faib ua ob peb kawm: zoo tib yam thiab ib nrab.

Nyob rau hauv no tsab xov xwm, peb yuav xav txog qhov zoo tib yam differential sib npaug ntawm cov thawj kev txiav txim. Piv txwv thiab cov kev daws peb sib tham txog nyob rau hauv cov nram qab no. Peb xav txog xwb lub TAC vim hais tias nws yog lub feem ntau hom kev sib npaug. Ordinary muab faib mus rau hauv subspecies: nrog separable variables, homogeneous thiab heterogeneous. Tom ntej no koj yuav kawm tau li cas lawv txawv los ntawm txhua lwm yam, thiab kev kawm yuav ua li cas los daws kom tau lawv.

Nyob rau hauv tas li ntawd, cov equations yuav muab tag nrho, yog li ntawd tom qab peb tau txais ib tug system ntawm differential sib npaug ntawm cov thawj kev txiav txim. Tej systems, peb kuj saib thiab kawm tau li cas los daws kom tau.

Vim li cas peb xav xwb cov thawj thiaj li? Vim hais tias nws yog tsim nyog los pib nrog ib tug yooj yooj yim thiab piav qhia txog tag nrho cov txuam nrog differential sib npaug, nyob rau hauv ib tsab xov xwm nws yog tsis yooj yim sua.

Sib npaug nrog separable variables

Qhov no yog tej zaum cov feem ntau yooj yim ua ntej kev txiav txim differential sib npaug. Cov no yog cov piv txwv uas yuav sau li: y '= f (x) * f (y). Yuav kom daws tau qhov no kab zauv peb xav tau tus sawv cev mis ntawm lub derivative li tus piv ntawm cov differentials: y '= dy / dx. Nrog nws peb muab tau cov kab zauv: dy / dx = f (x) * f (y). Tam sim no peb muaj peev xwm tig mus rau cov qauv ntawm cov kev daws txheem piv txwv: cais lub zog nyob rau hauv qhov chaw, piv txwv li ceev ceev rau pem hauv ntej rau tag nrho cov nce mus nce los y nyob rau hauv lub qhov chaw uas muaj yog dy, thiab kuj ua lub nce mus nce los x ... Peb tau txais ib kab zauv ntawm daim ntawv no: dy / f (y) = f (x) dx, uas yog tiav los ntawm kev noj cov integrals ntawm ob qhov chaw. Tsis txhob hnov qab txog lub qhov uas koj xav kom muab tso rau tom qab koom ua ke.

Cov tshuaj ntawm tej yam "diffura" - yog ib tug muaj nuj nqi ntawm x los ntawm y (nyob rau hauv peb cov ntaub ntawv), los yog yog hais tias muaj yog ib tug hais txog zauv mob, cov lus teb yog ib tug xov tooj. Cia peb kawm ib qhob piv txwv rau hauv tag nrho cov chav kawm ntawm qhov kev txiav txim:

y '= 2y * kev txhaum (x)

Hloov lub zog nyob rau hauv txawv kev qhia:

dy / y = 2 * kev txhaum (x) dx

Tam sim no siv lub integrals. Tag nrho cov ntawm lawv muaj peev xwm yuav nrhiav tau nyob rau hauv ib tug tshwj xeeb cov lus ntawm integrals. Thiab peb tau txais:

haujlwm uas yog COy (y) = -2 * cos (x) + C

Yog hais tias yuav tsum tau, peb yuav nthuav qhia qhov "y" raws li ib tug muaj nuj nqi ntawm "X". Tam sim no peb muaj peev xwm hais tias peb differential kab zauv solved, yog hais tias tsis teev mob. Yuav teev mob, piv txwv li, y (n / 2) = e. Tom qab ntawd peb yuav tsuas hloov tau tus nqi ntawm cov tsiaj ntawv nyob rau hauv qhov kev txiav txim thiab nrhiav tau tus nqi ntawm lub qhov. Nyob rau hauv peb tus yam ntxwv, nws yog 1.

Homogeneous thawj thiaj li differential sib npaug

Tam sim no mus rau lub ntau qhov chaw. Homogeneous thawj thiaj li differential sib npaug yuav sau nyob rau hauv kev daim ntawv raws li: y '= z (x, y). Nws yuav tsum tau muab sau tseg tias txoj kev muaj nuj nqi ntawm ob ntau yam yog uniform, thiab nws yuav tsis tau muab faib ua ob nyob rau hauv: z x thiab z ntawm y. Kos seb tus kab zauv homogeneous los yog tsis, yog heev yooj yim: peb ua lub hloov x = k * x thiab y = k * y. Tam sim no peb txiav tag nrho k. Yog hais tias cov ntawv no thiaj poob, ces tus kab zauv homogeneous thiab yuav xyuam xim mus rau nws cov tshuaj. Nrhiav ntej, peb hais tias: lub hauv paus ntsiab lus ntawm cov tshuaj ntawm cov piv txwv no kuj yooj yim heev.

Peb yuav tsum tau los mus ua kom cov hloov: y = t (x) * x, qhov twg t - ib tug muaj nuj nqi uas tseem nyob rau x. Tom qab ntawd peb yuav qhia cov derivative: y '= t' (x) * x + t. Pauv tag nrho cov no mus rau hauv peb thawj kab zauv thiab simplifying nws, peb muaj cov piv txwv ntawm qhov kev sib cais ntawm cov tsiaj ntawv t li x. Daws nws thiab muab tau cov nyob tos ntawm t (x). Thaum peb tau txais, tsuas hloov peb yav dhau los hloov y = t (x) * x. Tom qab ntawd peb tau lub dependence ntawm y rau x.

Kom nws dlaim phaj, peb yuav tsum to taub ib qho piv txwv: x * y '= yx * e y / x.

Thaum kuaj tau cov kev hloov ntawm tag nrho cov koos. Yog li ntawd, cov kab zauv tiag tiag homogeneous. Tam sim no ua kom lwm tus hloov, peb tham txog: y = t (x) * x thiab y '= t' (x) * x + t (x). Tom qab simplification nram qab no kab zauv: t '(x) * x = -e t. Peb txiav txim siab mus yuav tau ib tug qauv nrog sib cais ntau yam thiab peb ^{tau txais: e -t = ln (C *} x). Peb cia li yuav tsum tau hloov t los ntawm y / x (vim hais tias yog y = t * x, ces t = y / x), thiab peb tau txais cov ^{lus teb: e -y / x = ln (} x * C).

Tawm differential kab zauv ntawm thawj thiaj li

Nws yog lub sij hawm mus xav txog lwm sab lub npe. Peb yuav saib ua kev thawj txiav txim differential sib npaug. Yuav ua li cas lawv txawv los ntawm lub yav dhau los ob? Wb fim nws. Tawm thawj thiaj li differential sib npaug nyob rau hauv lub dav dav hauv daim ntawv ntawm cov kab zauv yuav sau li no: y '+ g (x) * y = z (x). Nws yuav tsum tau clarified tias z (x) thiab g (x) tej zaum yuav qhov tseem ceeb.

Ntawm no yog ib qho piv txwv: y '- y * x = x ^2.

Muaj ob txoj kev los daws kom tau, thiab peb txiav txim kom Cia peb kawm ob qho tib si ntawm lawv. Tus thawj - cov qauv ntawm cov variation ntawm arbitrary constants.

Yuav kom daws tau cov kab zauv nyob rau hauv no yav tas, nws yog tsim nyog los equate lub thawj txoj cai-hand rau zero, thiab daws tau qhov uas ua kab zauv uas tom qab cov kev hloov ntawm qhov chaw yuav:

y '= y * x;

dy / dx = y * x;

dy / y = xdx;

haujlwm uas yog COy | y | = x ^2/2 + C;

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

Tam sim no nws yog tsim nyog los hloov rau qhov C ₁ rau cov nuj nqi v (x), uas peb yuav nrhiav tau.

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

Kos ib tug hloov derivative:

^{y '= v' * e x2} / ² -hluav * v * e ^{x2 / 2.}

Thiab hloov cov kab zauv rau hauv tus thawj kab zauv:

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

Koj yuav pom tias nyob rau hauv rau sab laug sab ntawm lub ob nqe lus yog txo. Yog hais tias ib co piv txwv tias tsis tshwm sim, ces koj tau ua ib yam dab tsi tsis ncaj ncees lawm. Peb mus txuas ntxiv mus rau:

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

Tam sim no peb daws tau qhov ib txwm kab zauv nyob rau hauv uas koj xav kom cais lub zog:

DV / dx = x ^{2 /} e ^x2 / ^2;

DV = x ² * e ^{- x 2/2} dx.

Yuav kom tshem tawm cov tseem ceeb heev, peb yuav tau thov kev koom ua ke los ntawm qhov chaw no. Txawm li cas los, qhov no tsis yog lub npe ntawm no tsab xov xwm. Yog hais tias koj xav, koj yuav kawm tau nyob rau hauv lawv tus kheej mus nqa tawm xws li kev ua. Nws yog tsis yooj yim, thiab muaj txaus kev txawj thiab kev kho mob yog tsis sij hawm haus mas.

Xa mus rau lub thib ob txoj kev daws ntawm lub inhomogeneous sib npaug: Bernoulli txoj kev. Yuav ua li cas kom ze yog sai thiab yooj yim - nws yog rau koj.

Yog li ntawd, thaum daws cov qauv no, peb yuav tsum tau los mus ua kom cov hloov: y = k * n. Ntawm no, k thiab n - ib co zog nyob ntawm x. Ces tus derivative yuav zoo li: y '= k' * n + k * n '. Tus seev cev ntawm ob tug hloov nyob rau hauv kab zauv:

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

Group li:

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

Tam sim no nws yog tsim nyog los equate rau zero, uas yog nyob rau hauv koov. Tam sim no, Yog hais tias koj muab ob tug ua sib npaug, peb nrhiav tau ib tug system ntawm thawj thiaj li differential sib npaug yuav tsum tau solved:

n '+ x * n = 0;

k '* n = x ^2.

Tus thawj koob pheej ntawm lawv txiav txim siab yuav ua li cas lub ib txwm kab zauv. Ua li no, koj yuav tsum cais cov variables:

DN / dx = x * v;

DN / n = xdx.

Peb muab cov ib thiab peb tau: ln (n) = x ^2/2. Ces, Yog hais tias peb qhia n:

n = e ^{x2 / 2.}

Tam sim no tus lub resulting kab zauv rau hauv lub thib ob kab zauv:

k '* e ^{x2 / 2} = x ^2.

Thiab transforming, peb tau tib yam kab zauv raws li nyob rau hauv thawj txoj kev:

DK = x ^{2 /} e ^x2 / ^2.

Peb kuj yuav tsis tham ntxiv kev txiav txim. Nws yog hais tias thaum xub thawj ua ntej kev txiav txim differential sib npaug tshuaj ua loj teeb meem. Txawm li cas los, ib tug tob immersion nyob rau hauv lub npe pib tau zoo thiab zoo.

Qhov twg yog differential sib npaug?

Heev active differential sib npaug siv nyob rau hauv physics, raws li yuav luag tag nrho cov yooj yim cov kev cai no tau muab sau nyob rau hauv txawv daim ntawv no, thiab cov neeg qauv, uas peb pom - ib tug tov rau cov sib npaug. Nyob rau hauv Science News for KIDS, lawv yog siv rau cov tib yog vim li cas: lub yooj yim kev cai lij choj yog muab tau los ntawm lawv. Nyob rau hauv biology, lub differential sib npaug yog siv qauv tus cwj pwm ntawm systems, xws li predator - prey. Lawv kuj yuav siv tau los mus tsim cov qauv ntawm cov tu tub tu kiv, piv txwv li, zos ntawm microorganisms.

Raws li differential sib npaug pab nyob rau hauv lub neej?

Lo lus teb rau lo lus nug no yog yooj yim: tsis muaj dab tsi. Yog hais tias koj tsis yog ib tug paub txog los yog engineer, nws yog tsis zoo li hais tias lawv yuav pab tau. Txawm li cas los, tsis ua mob rau kom koj paub tias lub differential kab zauv thiab nws yog solved rau lub zuag qhia tag nrho kev loj hlob. Thiab ces rau lo lus nug ntawm ib tug tub los yog tus ntxhais, "cas ib tug txawv kab zauv?" tsis txhob muab koj nyob rau hauv ib tug tuag thaum xaus. Zoo, yog koj yog ib tug paub txog los yog engineer, ces koj paub tias qhov tseem ceeb ntawm no lub npe nyob rau hauv tej kev kawm. Tab sis tseem ceeb tshaj, uas tam sim no rau lo lus nug "yuav ua li cas los daws kom tau lub differential kab zauv ntawm thawj thiaj li?" koj yuav yeej ib txwm yuav tsum tau muab ib lo lus teb. Pom zoo, nws yog ib txwm zoo thaum koj twb paub lawm tias dab neeg yog txawm ntshai mus nrhiav tau tawm.

Lub ntsiab teeb meem nyob rau hauv txoj kev tshawb no

Cov ntsiab teeb meem nyob rau hauv qhov kev nkag siab ntawm no lub npe yog ib tug phem tus cwj pwm ntawm kev koom ua ke thiab ntau yam zog. Yog hais tias koj tsis xav hais tsis xav derivatives thiab integrals, nws yog tej zaum muaj nqis ntau rau kawm, kawm sib txawv txoj kev koom ua ke thiab ntau yam zoo, thiab xwb ces npaj mus rau txoj kev tshawb no ntawm cov ntaub ntawv uas tau piav nyob rau hauv tsab xov xwm.

Ib txhia neeg xav tsis thoob yuav paub tias dx yuav kis, raws li yav tas los (nyob rau hauv tsev kawm ntawv) sib cav hais tias lub feem dy / dx yog indivisible. Ces koj yuav tsum nyeem cov ntaub ntawv nyob rau hauv lub derivative thiab to taub hais tias nws yog tus cwj pwm ntawm infinitely me me qhov ntau, uas yuav tsum tau manipulated nyob rau hauv kev daws sib npaug.

Muaj coob tus neeg tsis tam sim ntawd twb paub lawm hais tias kev daws ntawm differential sib npaug ntawm thawj thiaj li - qhov no yog feem ntau ib tug muaj nuj nqi los yog neberuschiysya integral, thiab qhov no delusion muab lawv ib tug ntau ntawm teeb meem.

Yuav ua li cas lwm tus muaj peev xwm yuav kawm kom to taub zoo dua?

Nws yog qhov zoo tshaj plaws los pib ntxiv immersion rau hauv lub ntiaj teb no ntawm differential calculus tshwj xeeb cov phau ntawv, piv txwv li, nyob rau hauv lej tsom xam rau cov tub ntxhais kawm uas tsis yog-zauv specialties. Koj muaj peev xwm ces txav mus rau qhov ntau tshwj xeeb ntaub ntawv.

Nws yog tau hais tias, nyob rau hauv tas li ntawd mus rau lub txawv, muaj cov tseem ib sib npaug, yog li ntawd koj yuav yeej ib txwm muaj ib yam dab tsi kom siv zog rau thiab yuav ua li kawm.

xaus

Peb cia siab hais tias tom qab nyeem ntawv no tsab xov xwm koj yuav tau ib lub tswv yim dab tsi lub differential sib npaug thiab yuav ua li cas los daws kom tau lawv kom raug.

Nyob rau hauv txhua rooj plaub, kev ua lej nyob rau hauv txhua txoj kev pab tau rau peb nyob rau hauv lub neej. Nws muaj logic thiab saib xyuas, yam uas txhua txhua tus txiv neej, raws li tsis muaj tes.