differential equation of complex variable.
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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
asked Dec 14 '18 at 7:38
twinkling startwinkling star
817
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add a comment |
$begingroup$
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
asked Dec 14 '18 at 7:38
twinkling startwinkling star
817
$endgroup$
$begingroup$
Seems like $z=0$ is the only solution here. You have too many equations.
$endgroup$
– Dylan
Dec 14 '18 at 7:54
add a comment |
$begingroup$
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
asked Dec 14 '18 at 7:38
twinkling startwinkling star
817
$endgroup$
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
asked Dec 14 '18 at 7:38
twinkling startwinkling star
817
asked Dec 14 '18 at 7:38
twinkling startwinkling star
817
asked Dec 14 '18 at 7:38
twinkling startwinkling star
817
asked Dec 14 '18 at 7:38
twinkling startwinkling star
817
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
add a comment |
$begingroup$
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
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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 $0$.
answered Dec 14 '18 at 8:22
alex halex h
111
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add a comment |
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1 Answer
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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 $0$.
answered Dec 14 '18 at 8:22
alex halex h
111
$endgroup$
add a comment |
$begingroup$
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 $0$.
answered Dec 14 '18 at 8:22
alex halex h
111
$endgroup$
add a comment |
$begingroup$
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 $0$.
answered Dec 14 '18 at 8:22
alex halex h
111
$endgroup$
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 $0$.
answered Dec 14 '18 at 8:22
alex halex h
111
answered Dec 14 '18 at 8:22
alex halex h
111
answered Dec 14 '18 at 8:22
alex halex h
111
answered Dec 14 '18 at 8:22
alex halex h
111
111
add a comment |
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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