ࡱ> 6859 V+bjbj*Xl6 6 6 6 $ 4&&&&&&&$ &&&&&&&&j&&&& n 6 6 0]^ RWNANIA R{NICZKOWE JEDNORODNE WZGLDEM X i Y  EMBED Equation.3  stosujemy podstawienie  EMBED Equation.3  ==>  EMBED Equation.3  po zr|niczkowaniu  EMBED Equation.3  i po podstawieniu do rwnania  EMBED Equation.3  RWNANIA R{NICZKOWE TYPU  EMBED Equation.3  Przypadek I : a1b2  b1a2 ( 0 Wtedy ukBad rwnaD ma jedno rozwizanie x=( i y=( Wprowadzamy nowe zmienne : x - ( = u; y - ( = v Rwnanie r|niczkowe przeksztaBci si na :  EMBED Equation.3  Za v podstawiamy  EMBED Equation.3  =>  EMBED Equation.3  Rozdzielamy zmienne i caBkujemy. Przypadek II : a1b2  b1a2 = 0 wspBczynnik proporcjonalno[ci :  EMBED Equation.3  wprowadzamy now zmienn :  EMBED Equation.3  => po r|niczce  EMBED Equation.3  po uwzgldnieniu wzorw rwnanie przyjmie posta :  EMBED Equation.3  rozwizujemy metod rozdzielenia zmiennych RWNANIA R{NICZKOWE LINIOWE NIEJEDNORODNE  EMBED Equation.3  rozwizujemy tzw. metod uzmienniania staBej. Rozwizujemy najpierw rwnanie jednorodne  EMBED Equation.3  CaBk tego rwnania jest  EMBED Equation.3  Teraz staB C zastpujemy funkcj  EMBED Equation.3 i mamy  EMBED Equation.3  Obliczamy z tego pochodn :  EMBED Equation.3  podstawiamy, redukujemy i liczymy PRZYKAAD :  EMBED Equation.3  Rozwizujemy najpierw rwnanie jednorodne  EMBED Equation.3  (przez rozdzielenie zmiennych) Wynikiem jest  EMBED Equation.3  gdzie  EMBED Equation.3  uwalniajc si od logarytmw mamy  EMBED Equation.3  uzmienniamy staB C1 :  EMBED Equation.3  (**) obliczamy pochodn :  EMBED Equation.3  warto[ci y i y` wstawiamy do pierwszego rwnania EMBED Equation.3  skd po redukcji otrzymamy  EMBED Equation.3 czyli  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  podstawiamy to do rwnania (**) otrzymujc koDcowy wynik  EMBED Equation.3  RWNANIA R{NICZKOWE TYPU  EMBED Equation.3  Przewidzenie : y1 = meCx lub y1 = (mx+n)eCx PRZYKAAD  EMBED Equation.3  Przewidujemy :  EMBED Equation.3  Obliczamy z tego pochodn  EMBED Equation.3  Warto[ci  EMBED Equation.3 i  EMBED Equation.3 podstawiamy do pierwszego rwnania :  EMBED Equation.3  Skd po uproszczeniu przez e3x znajdujemy m=2 Podstawiajc m do rwnania na  EMBED Equation.3 otrzymujemy  EMBED Equation.3  Obliczamy teraz caBk ogln rwnania uproszczonego :  EMBED Equation.3  Po rozwizaniu mamy :  EMBED Equation.3  CaBka oglna caBego rwnania to y1+y2 czyli  EMBED Equation.3  RWNANIA R{NICZKOWE TYPU  EMBED Equation.3 gdzie a jest liczb staB, a Wn(x) jest wielomianem stopnia n PRZYKAAD:  EMBED Equation.3  Przewidzenie :  EMBED Equation.3  Obliczamy pochodn  EMBED Equation.3  Podstawiamy powy|sze warto[ci do pierwszego rwnania:  EMBED Equation.3  Przyrwnujc wspBczynniki przy tych samych potgach zmiennej x otrzymujemy zwizki: 2a=1; 2a+2b=0; b+2c=0; z tego mamy :  EMBED Equation.3  Rozwizujemy teraz rwnanie jednorodne  EMBED Equation.3  i z tego mamy  EMBED Equation.3  CaBka oglna ma wic posta y1 + y2 tj.  EMBED Equation.3  RWNANIA R{NICZKOWE TYPU  EMBED Equation.3  PRZEWIDZENIE :  EMBED Equation.3  PRZYKAAD  EMBED Equation.3  Przewidujemy :  EMBED Equation.3  Obliczamy z tego pochodn :  EMBED Equation.3  Wstawiamy to do pierwszej rwno[ci :  EMBED Equation.3  Obliczamy n i m i mamy  EMBED Equation.3  Obliczamy rwnanie jednorodne EMBED Equation.3  z czego mamy  EMBED Equation.3  No i caBka oglna ma posta :  EMBED Equation.3  RWNANIA R{NICZKOWE INNYCH TYPW PRZYKAAD 1  EMBED Equation.3  PRZEWIDZENIE :  EMBED Equation.3  Obliczamy pochodn  EMBED Equation.3  wstawiamy to do rwnania pierwszego i dzielimy od razu przez e3x, otrzymujemy :  EMBED Equation.3  znajdujemy a,b,c i mamy  EMBED Equation.3  Liczymy caBk ogln rwnania jednorodnego EMBED Equation.3  i mamy ostatecznie  EMBED Equation.3  PRZYKAAD 2  EMBED Equation.3  PRZEWIDUJEMY :  EMBED Equation.3  Liczymy pochodn :  EMBED Equation.3  Wstawiamy to do pierwszego rwnania i mamy rwnanie, obliczamy wspBczynniki  EMBED Equation.3  liczymy caBk ogln rwnania jednorodnego i ostatecznie mamy :  EMBED Equation.3  RWNANIA R{NICZKOWE RZDU DRUGIEGO TYP :  EMBED Equation.3  Podstawienie :  EMBED Equation.3  PRZYKAAD  EMBED Equation.3  Robimy podstawienie :  EMBED Equation.3  i otrzymujemy  EMBED Equation.3  rozdzielamy zmienne  EMBED Equation.3  skd mamy lnp = ln(1+x) +lnC, a wiec  EMBED Equation.3  to znaczy  EMBED Equation.3  czyli  EMBED Equation.3  ostatecznie otrzymujemy  EMBED Equation.3  TYP :  EMBED Equation.3  Podstawienie :  EMBED Equation.3  wwczas  EMBED Equation.3  a rwnanie przeksztaBci si na rwnanie rzdu pierwszego ^`"$&(PRxz|~$&LNPRTüufj 5B*EHU\phj7? 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[vPo,Ul]vvhnYVs`Ժ.y5GG$pvs/a۫Cۣ/CompObj')jfObjInfo*lEquation Native m_10606314535:-F`m `m Ole pCompObj,.qfObjInfo/sEquation Native t FRwnanie Microsoft 3.0 DS Equation Equation.39q7pI0oI y 1 =12sin3x"32cos3x_1060631399?2F`m `m Ole wCompObj13xfObjInfo4z FRwnanie Microsoft 3.0 DS Equation Equation.39q74:JpI dydx+y=0 FRwnanie Microsoft 3.0 DS Equation Equation.39qEquation Native {P_10606314127F`m `m Ole }CompObj68~fObjInfo9Equation Native T_1060631460<F`m `m Ole 78J/J y 2 =Ce "x FRwnanie Microsoft 3.0 DS Equation Equation.39q7|Equation Native _1060632089+IAF`m `m in3x"32cos3x+Ce "x FRwnanie Microsoft 3.0 DS Equation Equation.39q7doI {I dydx+2y=25x 2 e 3xOle CompObj@BfObjInfoCEquation Native _1060632139FFm m Ole CompObjEGfObjInfoH FRwnanie Microsoft 3.0 DS Equation Equation.39q7hIPI y 1 =(ax 2 +bx+c)e 3xEquation Native _1060632185DNKFm m Ole CompObjJLf FRwnanie Microsoft 3.0 DS Equation Equation.39q7ݸLII dy 1 dx=(2ax+b)e 3x +3(ax 2 +bx+c)e 3xObjInfoMEquation Native _1060632273PFm m Ole CompObjOQfObjInfoREquation Native _10606323470;UFm m  FRwnanie Microsoft 3.0 DS Equation Equation.39q7ݰI,I 2ax+b+3(ax 2 +bx+c)+2(ax 2 +bx+c)a"25x 2Ole CompObjTVfObjInfoWEquation Native  FRwnanie Microsoft 3.0 DS Equation Equation.39q7t`I?@ABCDEFGHIJKLOQRSTUVWXYZ[\]^_`uation Equation.39q784IhI dydx+2y=0 FRwnanie Microsoft 3.0 DS Equation Equation.39qEquation Native T_1060632425Xb_Fm m Ole CompObj^`fObjInfoaEquation Native _1060633047dFm m Ole 7݈ JI y=(5x 2 "2x+25)e 3x +Ce "2x FRwnanie Microsoft 3.0 DS Equation Equation.39q7HoI {I dydx+CompObjcefObjInfofEquation Native d_1060633085]1iFm m y=2xcosx1 FRwnanie Microsoft 3.0 DS Equation Equation.39q7tIPI y 1 =(ax+b)sinx+(cx+d)cosxOle CompObjhjfObjInfokEquation Native _1060633123nFm m Ole CompObjmofObjInfop FRwnanie Microsoft 3.0 DS Equation Equation.39q7ݸII dy 1 dx=asinx+(ax+b)cosx+ccosx"(cx+d)sinxEquation Native _1060633215lvsFm m Ole CompObjrtf FRwnanie Microsoft 3.0 DS Equation Equation.39q7\ĩII y 1 =(x"1)sinx+xcosx FRwnanie Microsoft 3.0 DS EqObjInfouEquation Native x_1060633260xFm m Ole CompObjwyfObjInfozEquation Native _1060700814@}Fm m uation Equation.39q7lII y=(x"1)sinx+xcosx+Ce "x FRwnanie Microsoft 3.0 DS Equation Equation.39qOle CompObj|~fObjInfoEquation Native \s@oI {I F(x,y`,y``)=0 FRwnanie Microsoft 3.0 DS Equation Equation.39qs(\I8I y`=p(x)_1060700890Fm m Ole CompObjfObjInfoEquation Native D_1060700917{Fm @m Ole CompObjf FRwnanie Microsoft 3.0 DS Equation Equation.39qs8IHI (1+x)y``=y`ObjInfoEquation Native T_1060700950F@m @m Ole CompObjfObjInfoEquation Native D_1060700974F@m @m  FRwnanie Microsoft 3.0 DS Equation Equation.39qs(\I8I y`=p(x) FRwnanie Microsoft 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native \s@4{II (1+x)dpdx=p FRwnanie Microsoft 3.0 DS Equation Equation.39qs@II dpp=_1060701004F@m @m Ole CompObjfObjInfoEquation Native \_1060701078F@m @m Ole CompObjf  !$'*+-./01245678:dx1+x FRwnanie Microsoft 3.0 DS Equation Equation.39qs,IlpI p=C(1+x)^ObjInfoEquation Native H_1060701096F@m @m Ole CompObjfObjInfo Equation Native  T_1060701121F@m @m  FRwnanie Microsoft 3.0 DS Equation Equation.39qs8II dydx=C+Cx FRwnanie Microsoft 3.0 DS Equation Equation.39qOle  CompObjfObjInfoEquation Native \s@II y=(Cx+C)dx  +" FRwnanie Microsoft 3.0 DS Equation Equation.39qs\JpI y=12C_1060701146F`Zm `Zm Ole CompObjfObjInfoEquation Native x_1060701241F`Zm `Zm Ole CompObjfx 2 +Cx+C 1 FRwnanie Microsoft 3.0 DS Equation Equation.39qs@HJI F(y,y`,y``)=0ObjInfoEquation Native \_1060701257F`Zm `Zm Ole CompObj fObjInfo"Equation Native #D_1060701293F`Zm `Zm  FRwnanie Microsoft 3.0 DS Equation Equation.39qs(II y`=p(y) FRwnanie Microsoft 3.0 DS Equation Equation.39qOle %CompObj&fObjInfo(Equation Native )sۜ0JI y``=dudx=dudy"dydx=dudyu(y) Oh+'0 $0 L X dpx/RWNANIA RӯNICZKOWE JEDNORODNE WZGLDEM X i Y0ik ΋;2+c_z\S(mVCocn#7F=\XŽ`ܣ%/wcnDd$J U C AT? 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