How do integrate equation with two derivatives?

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I need to integrate next equation:



$$m cdot dz=left( r-z(r-1) right)^4 cdot dfracdpdz cdot left( p+k+dfrack^2pright)dz$$



where $m$, $r$, $k$ are constants and $p=p(z)$. How to integrate this equation-its right side? Would be appropriate to cut terms $dz$ because of the term $r-z(r-1)$?




integration differential-equations derivatives indefinite-integrals




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    up vote
    0
    down vote

    favorite












    I need to integrate next equation:



    $$m cdot dz=left( r-z(r-1) right)^4 cdot dfracdpdz cdot left( p+k+dfrack^2pright)dz$$



    where $m$, $r$, $k$ are constants and $p=p(z)$. How to integrate this equation-its right side? Would be appropriate to cut terms $dz$ because of the term $r-z(r-1)$?




    integration differential-equations derivatives indefinite-integrals




    share|cite|improve this question






















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I need to integrate next equation:



      $$m cdot dz=left( r-z(r-1) right)^4 cdot dfracdpdz cdot left( p+k+dfrack^2pright)dz$$



      where $m$, $r$, $k$ are constants and $p=p(z)$. How to integrate this equation-its right side? Would be appropriate to cut terms $dz$ because of the term $r-z(r-1)$?




      integration differential-equations derivatives indefinite-integrals




      share|cite|improve this question












      I need to integrate next equation:



      $$m cdot dz=left( r-z(r-1) right)^4 cdot dfracdpdz cdot left( p+k+dfrack^2pright)dz$$



      where $m$, $r$, $k$ are constants and $p=p(z)$. How to integrate this equation-its right side? Would be appropriate to cut terms $dz$ because of the term $r-z(r-1)$?





      integration differential-equations derivatives indefinite-integrals





      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Aug 24 at 9:45









      nick_name

      1209




      1209




















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          $$m= left( r-z left( r-1 right) right) ^4 left( frac rm d
          rm dzp left( z right) right) left( p left( z right) +k+
          frac k^2p left( z right) right)$$



          $$m left( p left( z right) +k+frac k^2p left( z right)
          right) ^-1= left( r-z left( r-1 right) right) ^4frac
          rm drm dzp left( z right)$$



          $$frac m left( r-z left( r-1 right) right) ^4= left( p
          left( z right) +k+frac k^2p left( z right) right)
          frac rm drm dzp left( z right)$$



          $$int !frac m left( r-z left( r-1 right) right) ^4
          ,rm dz=int !p(z)+k+frac k^2p(z),rm dp(z)
          $$



          $$-1/3,frac m left( left( 1-r right) z+r right) ^3 left( 1
          -r right) =1/2,p(z)^2+kp(z)+k^2ln left( p(z) right) +C
          $$






          share|cite|improve this answer






















          • Thanks, it works, but is the result of $int dfracdz(r-z(r-z))^4$ that what you wrote, or $dfrac13(r-1)(r-z(r-z))^3$ ?
            – nick_name
            Aug 24 at 10:50











          • It is as it need to be now.
            – nick_name
            Aug 24 at 10:51










          Your Answer




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          $$m= left( r-z left( r-1 right) right) ^4 left( frac rm d
          rm dzp left( z right) right) left( p left( z right) +k+
          frac k^2p left( z right) right)$$



          $$m left( p left( z right) +k+frac k^2p left( z right)
          right) ^-1= left( r-z left( r-1 right) right) ^4frac
          rm drm dzp left( z right)$$



          $$frac m left( r-z left( r-1 right) right) ^4= left( p
          left( z right) +k+frac k^2p left( z right) right)
          frac rm drm dzp left( z right)$$



          $$int !frac m left( r-z left( r-1 right) right) ^4
          ,rm dz=int !p(z)+k+frac k^2p(z),rm dp(z)
          $$



          $$-1/3,frac m left( left( 1-r right) z+r right) ^3 left( 1
          -r right) =1/2,p(z)^2+kp(z)+k^2ln left( p(z) right) +C
          $$






          share|cite|improve this answer






















          • Thanks, it works, but is the result of $int dfracdz(r-z(r-z))^4$ that what you wrote, or $dfrac13(r-1)(r-z(r-z))^3$ ?
            – nick_name
            Aug 24 at 10:50











          • It is as it need to be now.
            – nick_name
            Aug 24 at 10:51














          up vote
          1
          down vote



          accepted










          $$m= left( r-z left( r-1 right) right) ^4 left( frac rm d
          rm dzp left( z right) right) left( p left( z right) +k+
          frac k^2p left( z right) right)$$



          $$m left( p left( z right) +k+frac k^2p left( z right)
          right) ^-1= left( r-z left( r-1 right) right) ^4frac
          rm drm dzp left( z right)$$



          $$frac m left( r-z left( r-1 right) right) ^4= left( p
          left( z right) +k+frac k^2p left( z right) right)
          frac rm drm dzp left( z right)$$



          $$int !frac m left( r-z left( r-1 right) right) ^4
          ,rm dz=int !p(z)+k+frac k^2p(z),rm dp(z)
          $$



          $$-1/3,frac m left( left( 1-r right) z+r right) ^3 left( 1
          -r right) =1/2,p(z)^2+kp(z)+k^2ln left( p(z) right) +C
          $$






          share|cite|improve this answer






















          • Thanks, it works, but is the result of $int dfracdz(r-z(r-z))^4$ that what you wrote, or $dfrac13(r-1)(r-z(r-z))^3$ ?
            – nick_name
            Aug 24 at 10:50











          • It is as it need to be now.
            – nick_name
            Aug 24 at 10:51












          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          $$m= left( r-z left( r-1 right) right) ^4 left( frac rm d
          rm dzp left( z right) right) left( p left( z right) +k+
          frac k^2p left( z right) right)$$



          $$m left( p left( z right) +k+frac k^2p left( z right)
          right) ^-1= left( r-z left( r-1 right) right) ^4frac
          rm drm dzp left( z right)$$



          $$frac m left( r-z left( r-1 right) right) ^4= left( p
          left( z right) +k+frac k^2p left( z right) right)
          frac rm drm dzp left( z right)$$



          $$int !frac m left( r-z left( r-1 right) right) ^4
          ,rm dz=int !p(z)+k+frac k^2p(z),rm dp(z)
          $$



          $$-1/3,frac m left( left( 1-r right) z+r right) ^3 left( 1
          -r right) =1/2,p(z)^2+kp(z)+k^2ln left( p(z) right) +C
          $$






          share|cite|improve this answer














          $$m= left( r-z left( r-1 right) right) ^4 left( frac rm d
          rm dzp left( z right) right) left( p left( z right) +k+
          frac k^2p left( z right) right)$$



          $$m left( p left( z right) +k+frac k^2p left( z right)
          right) ^-1= left( r-z left( r-1 right) right) ^4frac
          rm drm dzp left( z right)$$



          $$frac m left( r-z left( r-1 right) right) ^4= left( p
          left( z right) +k+frac k^2p left( z right) right)
          frac rm drm dzp left( z right)$$



          $$int !frac m left( r-z left( r-1 right) right) ^4
          ,rm dz=int !p(z)+k+frac k^2p(z),rm dp(z)
          $$



          $$-1/3,frac m left( left( 1-r right) z+r right) ^3 left( 1
          -r right) =1/2,p(z)^2+kp(z)+k^2ln left( p(z) right) +C
          $$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Aug 24 at 10:53

























          answered Aug 24 at 10:35









          Mariusz Iwaniuk

          1,7211615




          1,7211615











          • Thanks, it works, but is the result of $int dfracdz(r-z(r-z))^4$ that what you wrote, or $dfrac13(r-1)(r-z(r-z))^3$ ?
            – nick_name
            Aug 24 at 10:50











          • It is as it need to be now.
            – nick_name
            Aug 24 at 10:51
















          • Thanks, it works, but is the result of $int dfracdz(r-z(r-z))^4$ that what you wrote, or $dfrac13(r-1)(r-z(r-z))^3$ ?
            – nick_name
            Aug 24 at 10:50











          • It is as it need to be now.
            – nick_name
            Aug 24 at 10:51















          Thanks, it works, but is the result of $int dfracdz(r-z(r-z))^4$ that what you wrote, or $dfrac13(r-1)(r-z(r-z))^3$ ?
          – nick_name
          Aug 24 at 10:50





          Thanks, it works, but is the result of $int dfracdz(r-z(r-z))^4$ that what you wrote, or $dfrac13(r-1)(r-z(r-z))^3$ ?
          – nick_name
          Aug 24 at 10:50













          It is as it need to be now.
          – nick_name
          Aug 24 at 10:51




          It is as it need to be now.
          – nick_name
          Aug 24 at 10:51

















           

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