differential equation of complex variable.












1












$begingroup$


I want to solve the differential equation with two complex variables.
$$ frac{partial^2 z}{partial w_1^2} = -z ;,; frac{partial^2 z}{partial w_2^2} = -z$$
$$ frac{partial^2 z}{partial w_1partial w_2} = frac{partial^2 z}{partial w_2partial w_1}=0$$
where $z(w_1, w_2) , w_1, w_2$ are complex variables.



I tried to use the case of solving equation $y=y(x) $ with $y'' +y =0 $, but I have no idea for multi-variable case.



How should I solve this?






ordinary-differential-equations







share|cite|improve this question









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  • $begingroup$
    Seems like $z=0$ is the only solution here. You have too many equations.
    $endgroup$
    – Dylan
    Dec 14 '18 at 7:54
















1












$begingroup$


I want to solve the differential equation with two complex variables.
$$ frac{partial^2 z}{partial w_1^2} = -z ;,; frac{partial^2 z}{partial w_2^2} = -z$$
$$ frac{partial^2 z}{partial w_1partial w_2} = frac{partial^2 z}{partial w_2partial w_1}=0$$
where $z(w_1, w_2) , w_1, w_2$ are complex variables.



I tried to use the case of solving equation $y=y(x) $ with $y'' +y =0 $, but I have no idea for multi-variable case.



How should I solve this?






ordinary-differential-equations







share|cite|improve this question









$endgroup$












  • $begingroup$
    Seems like $z=0$ is the only solution here. You have too many equations.
    $endgroup$
    – Dylan
    Dec 14 '18 at 7:54














1












1








1


1



$begingroup$


I want to solve the differential equation with two complex variables.
$$ frac{partial^2 z}{partial w_1^2} = -z ;,; frac{partial^2 z}{partial w_2^2} = -z$$
$$ frac{partial^2 z}{partial w_1partial w_2} = frac{partial^2 z}{partial w_2partial w_1}=0$$
where $z(w_1, w_2) , w_1, w_2$ are complex variables.



I tried to use the case of solving equation $y=y(x) $ with $y'' +y =0 $, but I have no idea for multi-variable case.



How should I solve this?






ordinary-differential-equations







share|cite|improve this question









$endgroup$




I want to solve the differential equation with two complex variables.
$$ frac{partial^2 z}{partial w_1^2} = -z ;,; frac{partial^2 z}{partial w_2^2} = -z$$
$$ frac{partial^2 z}{partial w_1partial w_2} = frac{partial^2 z}{partial w_2partial w_1}=0$$
where $z(w_1, w_2) , w_1, w_2$ are complex variables.



I tried to use the case of solving equation $y=y(x) $ with $y'' +y =0 $, but I have no idea for multi-variable case.



How should I solve this?






ordinary-differential-equations




ordinary-differential-equations






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 14 '18 at 7:38









twinkling startwinkling star

817




817












  • $begingroup$
    Seems like $z=0$ is the only solution here. You have too many equations.
    $endgroup$
    – Dylan
    Dec 14 '18 at 7:54


















  • $begingroup$
    Seems like $z=0$ is the only solution here. You have too many equations.
    $endgroup$
    – Dylan
    Dec 14 '18 at 7:54
















$begingroup$
Seems like $z=0$ is the only solution here. You have too many equations.
$endgroup$
– Dylan
Dec 14 '18 at 7:54




$begingroup$
Seems like $z=0$ is the only solution here. You have too many equations.
$endgroup$
– Dylan
Dec 14 '18 at 7:54










1 Answer
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$begingroup$

Try starting with your bottom two equations.



(this is assuming that $z, w_1, w_2$ are all complex)



Letting $frac{partial z}{partial w_1} = F(z)$ and $frac{partial z}{partial w_2} = G(z)$, then you should note that $frac{partial z}{partial w_1 partial w_2} = frac{partial F}{partial w_2} = 0 =frac{partial G}{partial w_1} = frac{partial z}{partial w_2 partial w_1}$.



Therefore, $F(z)$ and $G(z)$ must be constants.



Letting some $z_o$ stand for your constant, you can then plug into the upper two equations. Note that the only constant for which your second partial derivative is $-z$, is $0$.






share|cite|improve this answer









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    $begingroup$

    Try starting with your bottom two equations.



    (this is assuming that $z, w_1, w_2$ are all complex)



    Letting $frac{partial z}{partial w_1} = F(z)$ and $frac{partial z}{partial w_2} = G(z)$, then you should note that $frac{partial z}{partial w_1 partial w_2} = frac{partial F}{partial w_2} = 0 =frac{partial G}{partial w_1} = frac{partial z}{partial w_2 partial w_1}$.



    Therefore, $F(z)$ and $G(z)$ must be constants.



    Letting some $z_o$ stand for your constant, you can then plug into the upper two equations. Note that the only constant for which your second partial derivative is $-z$, is $0$.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      Try starting with your bottom two equations.



      (this is assuming that $z, w_1, w_2$ are all complex)



      Letting $frac{partial z}{partial w_1} = F(z)$ and $frac{partial z}{partial w_2} = G(z)$, then you should note that $frac{partial z}{partial w_1 partial w_2} = frac{partial F}{partial w_2} = 0 =frac{partial G}{partial w_1} = frac{partial z}{partial w_2 partial w_1}$.



      Therefore, $F(z)$ and $G(z)$ must be constants.



      Letting some $z_o$ stand for your constant, you can then plug into the upper two equations. Note that the only constant for which your second partial derivative is $-z$, is $0$.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        Try starting with your bottom two equations.



        (this is assuming that $z, w_1, w_2$ are all complex)



        Letting $frac{partial z}{partial w_1} = F(z)$ and $frac{partial z}{partial w_2} = G(z)$, then you should note that $frac{partial z}{partial w_1 partial w_2} = frac{partial F}{partial w_2} = 0 =frac{partial G}{partial w_1} = frac{partial z}{partial w_2 partial w_1}$.



        Therefore, $F(z)$ and $G(z)$ must be constants.



        Letting some $z_o$ stand for your constant, you can then plug into the upper two equations. Note that the only constant for which your second partial derivative is $-z$, is $0$.






        share|cite|improve this answer









        $endgroup$



        Try starting with your bottom two equations.



        (this is assuming that $z, w_1, w_2$ are all complex)



        Letting $frac{partial z}{partial w_1} = F(z)$ and $frac{partial z}{partial w_2} = G(z)$, then you should note that $frac{partial z}{partial w_1 partial w_2} = frac{partial F}{partial w_2} = 0 =frac{partial G}{partial w_1} = frac{partial z}{partial w_2 partial w_1}$.



        Therefore, $F(z)$ and $G(z)$ must be constants.



        Letting some $z_o$ stand for your constant, you can then plug into the upper two equations. Note that the only constant for which your second partial derivative is $-z$, is $0$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 14 '18 at 8:22









        alex halex h

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