How do I study the domain of a Cauchy's problem without solving it?












0












$begingroup$


I have some problems that require to study the domain of the Cauchy's problem solution but I don't really know how to do that. For example,



$begin{cases}
y'=(y-sin x)^2+1+cos x\
y(0)=0
end{cases}$



I did few theorems about Cauchy's problem but none of them says where the solution is defined.






ordinary-differential-equations cauchy-problem







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    I have some problems that require to study the domain of the Cauchy's problem solution but I don't really know how to do that. For example,



    $begin{cases}
    y'=(y-sin x)^2+1+cos x\
    y(0)=0
    end{cases}$



    I did few theorems about Cauchy's problem but none of them says where the solution is defined.






    ordinary-differential-equations cauchy-problem







    share|cite|improve this question











    $endgroup$















      0












      0








      0


      1



      $begingroup$


      I have some problems that require to study the domain of the Cauchy's problem solution but I don't really know how to do that. For example,



      $begin{cases}
      y'=(y-sin x)^2+1+cos x\
      y(0)=0
      end{cases}$



      I did few theorems about Cauchy's problem but none of them says where the solution is defined.






      ordinary-differential-equations cauchy-problem







      share|cite|improve this question











      $endgroup$




      I have some problems that require to study the domain of the Cauchy's problem solution but I don't really know how to do that. For example,



      $begin{cases}
      y'=(y-sin x)^2+1+cos x\
      y(0)=0
      end{cases}$



      I did few theorems about Cauchy's problem but none of them says where the solution is defined.






      ordinary-differential-equations cauchy-problem




      ordinary-differential-equations cauchy-problem






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 16 '18 at 11:20









      Rebellos

      14.6k31247




      14.6k31247










      asked Dec 16 '18 at 11:08









      ArchimedessArchimedess

      236




      236






















          1 Answer
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          active

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          $begingroup$

          Set $u=y-sin x$, the the ODE reduces to
          $$
          u'=u^2+1.
          $$
          This has an easy solution via separation of variables with the then obvious maximal domain.



          For problems where such easy solutions are not possible, see for instance




          • Existence of solution $y(x)$ with $x in [0,frac12]$

          • Riccati D.E., vertical asymptotes

          • Prove that the IVP $begin{cases}dot{x}=x^3+e^{-t^2}\x(0)=1end{cases}$ has an unique solution defined on $I=(-1/9,1/9)$






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Ok, thank you.. I don't think my course is that advance but still interesting.
            $endgroup$
            – Archimedess
            Dec 16 '18 at 11:35











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          1 Answer
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          1 Answer
          1






          active

          oldest

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          active

          oldest

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          active

          oldest

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          2












          $begingroup$

          Set $u=y-sin x$, the the ODE reduces to
          $$
          u'=u^2+1.
          $$
          This has an easy solution via separation of variables with the then obvious maximal domain.



          For problems where such easy solutions are not possible, see for instance




          • Existence of solution $y(x)$ with $x in [0,frac12]$

          • Riccati D.E., vertical asymptotes

          • Prove that the IVP $begin{cases}dot{x}=x^3+e^{-t^2}\x(0)=1end{cases}$ has an unique solution defined on $I=(-1/9,1/9)$






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Ok, thank you.. I don't think my course is that advance but still interesting.
            $endgroup$
            – Archimedess
            Dec 16 '18 at 11:35
















          2












          $begingroup$

          Set $u=y-sin x$, the the ODE reduces to
          $$
          u'=u^2+1.
          $$
          This has an easy solution via separation of variables with the then obvious maximal domain.



          For problems where such easy solutions are not possible, see for instance




          • Existence of solution $y(x)$ with $x in [0,frac12]$

          • Riccati D.E., vertical asymptotes

          • Prove that the IVP $begin{cases}dot{x}=x^3+e^{-t^2}\x(0)=1end{cases}$ has an unique solution defined on $I=(-1/9,1/9)$






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Ok, thank you.. I don't think my course is that advance but still interesting.
            $endgroup$
            – Archimedess
            Dec 16 '18 at 11:35














          2












          2








          2





          $begingroup$

          Set $u=y-sin x$, the the ODE reduces to
          $$
          u'=u^2+1.
          $$
          This has an easy solution via separation of variables with the then obvious maximal domain.



          For problems where such easy solutions are not possible, see for instance




          • Existence of solution $y(x)$ with $x in [0,frac12]$

          • Riccati D.E., vertical asymptotes

          • Prove that the IVP $begin{cases}dot{x}=x^3+e^{-t^2}\x(0)=1end{cases}$ has an unique solution defined on $I=(-1/9,1/9)$






          share|cite|improve this answer











          $endgroup$



          Set $u=y-sin x$, the the ODE reduces to
          $$
          u'=u^2+1.
          $$
          This has an easy solution via separation of variables with the then obvious maximal domain.



          For problems where such easy solutions are not possible, see for instance




          • Existence of solution $y(x)$ with $x in [0,frac12]$

          • Riccati D.E., vertical asymptotes

          • Prove that the IVP $begin{cases}dot{x}=x^3+e^{-t^2}\x(0)=1end{cases}$ has an unique solution defined on $I=(-1/9,1/9)$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 16 '18 at 11:37

























          answered Dec 16 '18 at 11:16









          LutzLLutzL

          58.3k42054




          58.3k42054












          • $begingroup$
            Ok, thank you.. I don't think my course is that advance but still interesting.
            $endgroup$
            – Archimedess
            Dec 16 '18 at 11:35


















          • $begingroup$
            Ok, thank you.. I don't think my course is that advance but still interesting.
            $endgroup$
            – Archimedess
            Dec 16 '18 at 11:35
















          $begingroup$
          Ok, thank you.. I don't think my course is that advance but still interesting.
          $endgroup$
          – Archimedess
          Dec 16 '18 at 11:35




          $begingroup$
          Ok, thank you.. I don't think my course is that advance but still interesting.
          $endgroup$
          – Archimedess
          Dec 16 '18 at 11:35


















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