In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form
where n {\displaystyle n} is a real number. Some authors allow any real n {\displaystyle n} ,[1][2] whereas others require that n {\displaystyle n} not be 0 or 1.[3][4] The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named. The earliest solution, however, was offered by Gottfried Leibniz, who published his result in the same year and whose method is the one still used today.[5]
Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A notable special case of the Bernoulli equation is the logistic differential equation.
When n = 0 {\displaystyle n=0} , the differential equation is linear. When n = 1 {\displaystyle n=1} , it is separable. In these cases, standard techniques for solving equations of those forms can be applied. For n ≠ ≠ --> 0 {\displaystyle n\neq 0} and n ≠ ≠ --> 1 {\displaystyle n\neq 1} , the substitution u = y 1 − − --> n {\displaystyle u=y^{1-n}} reduces any Bernoulli equation to a linear differential equation
For example, in the case n = 2 {\displaystyle n=2} , making the substitution u = y − − --> 1 {\displaystyle u=y^{-1}} in the differential equation d y d x + 1 x y = x y 2 {\displaystyle {\frac {dy}{dx}}+{\frac {1}{x}}y=xy^{2}} produces the equation d u d x − − --> 1 x u = − − --> x {\displaystyle {\frac {du}{dx}}-{\frac {1}{x}}u=-x} , which is a linear differential equation.
Let x 0 ∈ ∈ --> ( a , b ) {\displaystyle x_{0}\in (a,b)} and
be a solution of the linear differential equation
Then we have that y ( x ) := [ z ( x ) ] 1 / ( 1 − − --> α α --> ) {\displaystyle y(x):=[z(x)]^{1/(1-\alpha )}} is a solution of
And for every such differential equation, for all α α --> > 0 {\displaystyle \alpha >0} we have y ≡ ≡ --> 0 {\displaystyle y\equiv 0} as solution for y 0 = 0 {\displaystyle y_{0}=0} .
Consider the Bernoulli equation
(in this case, more specifically a Riccati equation). The constant function y = 0 {\displaystyle y=0} is a solution. Division by y 2 {\displaystyle y^{2}} yields
Changing variables gives the equations
which can be solved using the integrating factor
Multiplying by M ( x ) {\displaystyle M(x)} ,
The left side can be represented as the derivative of u x 2 {\displaystyle ux^{2}} by reversing the product rule. Applying the chain rule and integrating both sides with respect to x {\displaystyle x} results in the equations
The solution for y {\displaystyle y} is
هذه المقالة عن موضوع الإدراك الحسي. لالإدراك بمعناه العقلي، طالع استعراف. الإدراك الحسي[1][2] (باللاتينية: Perceptio) بمعنى التصور أو التجميع أو الاستلام هو تنظيم المعلومات الحسية وتحديدها وتفسيرها من أجل تمثيل وفهم المعلومات أو البيئة المقدمة.[3] إدراك حسي معلو
Overview of the role of Christmas in Ukraine Christmas in UkraineTwelve-dish Christmas Eve supper by Jacques HnizdovskyAlso calledUkrainian: Різдво, RizdvoObserved byChristians, many non-ChristiansTypeChristian, culturalSignificanceCommemoration of the nativity of JesusCelebrationsGift-giving, family and other social gatherings, symbolic decoration, feasting, etc.ObservancesChurch servicesDateDecember 25: the Orthodox Church of Ukraine, Latin Church, the Ukrainian Greek Catholic Churc…
Twee cilindrische en twee schijfvormige implantaten in de bovenkaak Implantologie is het snelstgroeiende onderdeel van de tandheelkunde en heeft tot doel een of meerdere tanden te vervangen door tandimplantaten. Implantaten zijn kunstmatige wortels, meestal vervaardigd uit Titanium, die in het bot geplaatst worden. Op het moment dat zowel de behandelaar als de patiënt met de rug tegen de muur staan als het gaat om tandheelkundige oplossingen, biedt implantologie mogelijkheden die vroeger niet b…
Опис файлу Опис Обкладинка альбому Can't Be Tamed Джерело https://en.wikipedia.org/wiki/File:Miley_Cyrus_-_Can't_Be_Tamed.png Час створення 2010 Автор зображення Майлі Сайрус Ліцензія див. нижче Обґрунтування добропорядного використання для статті «Can't Be Tamed» [?] Мета використання Ілюстрація предмету ст…
Provincias de Angola Províncias de Angola País Angola AngolaCategoría Primer nivel de división administrativaDatos estadísticosNúmero actual 18Subdivisión(es) Municipios[editar datos en Wikidata] Las provincias de Angola (en portugués: província) son las subdivisiones administrativas de primer nivel del país. Un total de dieciocho provincias están organizadas[nota 1] en el territorio angoleño, cada provincia está organizada en partes más pequeñas que son los …
For the second book of the Inheritance trilogy, see The Broken Kingdoms. 4th episode of the 5th season of Once Upon a Time The Broken KingdomOnce Upon a Time episodeEpisode no.Season 5Episode 4Directed byAlrick RileyWritten byDavid H. Goodman & Jerome SchwartzProduction code504Original air dateOctober 18, 2015 (2015-10-18)Guest appearances Liam Garrigan as King Arthur Amy Manson as Merida Sinqua Walls as Lancelot Andrew Jenkins as Percival Joana Metrass as Queen Guinevere…
PurwantoWakil Dekan Fakultas Strategi Pertahanan/UnhanPetahanaMulai menjabat 2020 Informasi pribadiLahir23 Juni 1964 (umur 59)Madiun, Jawa TimurSuami/istriNy. Yuniva KurniawatiAlma materAkademi Angkatan Laut (1988)STIE Wilwatikta (2004)Universitas Krisnadwipayana (2012)Universitas Pertahanan Indonesia (2013)Karier militerPihak IndonesiaDinas/cabang TNI Angkatan LautMasa dinas1988—2023Pangkat Laksamana Muda TNISatuanKorps Pelaut (Kopaska)Sunting kotak info • L •…
1952 song by Hank Williams Jambalaya (On the Bayou)Single by Hank Williams With His Drifting CowboysB-sideWindow ShoppingPublishedJuly 28, 1952 (1952-07-28) Acuff-Rose Publications[1]ReleasedJuly 19, 1952 (1952-07-19)RecordedJune 13, 1952 (1952-06-13)[2]StudioCastle Studio, NashvilleGenreCountry & western, honky-tonk, country bluesLength2:52LabelMGM K 11283Songwriter(s) Hank Williams Moon Mullican Hank Williams With His Drifting Co…
Nagpur Metro's Aqua Line metro station The topic of this article may not meet Wikipedia's notability guideline for geographic features. Please help to demonstrate the notability of the topic by citing reliable secondary sources that are independent of the topic and provide significant coverage of it beyond a mere trivial mention. If notability cannot be shown, the article is likely to be merged, redirected, or deleted.Find sources: Shankar Nagar Square metro station – news …
2005 studio album by Jackson and His ComputerbandSmashStudio album by Jackson and His ComputerbandReleasedSeptember 19, 2005 (2005-09-19)Length66:34LabelWarpProducerJackson and His ComputerbandJackson and His Computerband chronology Smash(2005) Glow(2013) Alternative coverPromo sampler CD cover Professional ratingsAggregate scoresSourceRatingMetacritic66/100[1]Review scoresSourceRatingAllMusic[2]Exclaim!favorable[3]Pitchfork7.8/10[4]Stylus M…
Administrateur de la santé publique des États-Unis Sceau du Service de la santé publique des États-Unis (1798). Titulaire actuelVivek Murthydepuis le 25 mars 2021 Création 1798 Titre Vice-amiral 3 étoiles Mandant Président des États-Unis Durée du mandat 4 ans Premier titulaire John Maynard Woodworth (Supervising Surgeon) Résidence officielle Bureau du secrétaire adjoint à la Santé Site internet www.hhs.gov/surgeongeneral modifier L'administrateur de la santé publique des …
Australian climate change organisation Climate CouncilClimate Council logoPredecessorClimate CommissionFormationAustralia, 2013TypeNonprofit organizationPurposeAction and communications on climate change impacts and solutionsRegion served AustraliaOfficial language EnglishLeaderTim FlanneryCEOAmanda McKenzieKey peopleTim Flannery, Will Steffen, Gerry Hueston, Greg Mullins, Lesley Ann HughesWebsiteOfficial website The Climate Council is Australia's leading climate change communications non-profit…
Former municipality in Oppland, Norway Former municipality in Oppland, NorwayBiri Municipality Biri herredFormer municipalityView of the village of Biri Oppland within NorwayBiri within OpplandCoordinates: 60°57′20″N 10°36′41″E / 60.95558°N 10.6115°E / 60.95558; 10.6115CountryNorwayCountyOpplandDistrictVestopplandEstablished1 Jan 1838 • Created asFormannskapsdistriktDisestablished1 Jan 1964 • Succeeded byGjøvik MunicipalityAdministrati…
2014 book edited by American author Susan Kuklin Beyond Magenta AuthorSusan KuklinCountryUnited StatesLanguageEnglishGenreYoung adult non-fictionPublisherCandlewick PressPublication dateFebruary 11, 2014Pages192ISBN978-0-7636-5611-9 Beyond Magenta: Transgender Teens Speak Out is a 2014 book written by American author Susan Kuklin. For the book, Kuklin met and interviewed six transgender or gender-neutral young adults, describing their sense of identity before, during, and after transitioning. Be…
Arguments against the economic system of capitalism This article is about criticism of capitalism. For the political movement opposed to capitalism, see Anti-capitalism. Karl Marx's three volume Capital. A Critique of Political Economy is highly regarded as one of the greatest written critiques of capitalism. Criticism of capitalism is a critique of political economy that involves the rejection of, or dissatisfaction with the economic system of capitalism and its outcomes. Criticisms typically r…
Faisal IIRaja IrakBerkuasa4 April 1939 – 14 Juli 1958(19 tahun, 101 hari)PendahuluGhazi IPenerusMuhammad Najib ar-Ruba'i (sebagai Presiden Irak)WaliAbd al-Ilah dari HejazInformasi pribadiPemakamanRoyal Mausoleum, AdhamiyahWangsaHashemiteAyahGhazi IIbuPutri Aliya binti AliAgamaIslam Sunni[1] Faishal II GCVO (Arab: الملك فيصل الثاني; 2 Mei 1935 – 14 Juli 1958) ialah Raja Irak yang memerintah antara tahun 1939 hingga 1958. Faishal berusia 3 tah…
The Sinking of the Titanic is a work by British minimalist composer Gavin Bryars. Inspired by the story that the band on the RMS Titanic continued to perform as the ship sank in 1912, it imagines how the music performed by the band would reverberate through the water some time after they ceased performing. Composed between 1969 and 1972, the work is now considered one of the classics of British classical experimental music.[1] History Bryars' inspiration for the work came from a report t…
لمعانٍ أخرى، طالع نوية (توضيح). مُخططٌ للنواة، وبداخلها النوية. علم الأحياء الخلويالخلية الحيوانيةمكونات الخلية الحيوانية النموذجية: النوية النواة الريبوسوم (نقاط صغيرة) الحويصلة الشبكة الإندوبلازمية الخشنة جهاز غولجي (أو جسيم غولجي) الهيكل الخلوي الشبكة الإندوبلاز…
Legislative Assembly constituency in Goa State, India MaemConstituency No. 16 for the Goa Legislative AssemblyConstituency detailsCountryIndiaRegionWestern IndiaStateGoaDistrictNorth GoaLS constituencyNorth GoaTotal electors28,919[1]ReservationNoneMember of Legislative Assembly8th Goa Legislative AssemblyIncumbent Premendra Shet PartyBharatiya Janata Party Maem Assembly constituency is one of the 40 Goa Legislative Assembly constituencies of the state of Goa in southern India. Maem is al…
Minnesota AtheistsFormation1991 (1991) Minnesota, U.S.Type501(c)(3)PurposePromotion of atheism, secular humanism, educationLocationSt. Paul/Minneapolis, MinnesotaPresidentBen BlanchardWebsitewww.mnatheists.org Minnesota Atheists is a 501(c)(3) non-profit organization that seeks to promote the positive contributions of atheism and to maintain the separation of state and church[1] and is the largest atheist organization in the state of Minnesota. It is affiliated with Atheist Alliance…
Lokasi Pengunjung: 18.217.200.13