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What differential equation corresponds to this vector field?

2

$begingroup$

Here is a vector field:$$ vec F(x,y)=sin(x),sin(y), $$

where $x,y in (0,pi).$

How do you find the differential equation, that when solved gives the integral curves for this vector field? I made this plot on WolframAlpha:

ordinary-differential-equations multivariable-calculus vector-fields

share|cite|improve this question

asked Mar 24 at 16:34

UltradarkUltradark

3481518

$endgroup$

add a comment | 

2

$begingroup$

Here is a vector field:$$ vec F(x,y)=sin(x),sin(y), $$

where $x,y in (0,pi).$

How do you find the differential equation, that when solved gives the integral curves for this vector field? I made this plot on WolframAlpha:

ordinary-differential-equations multivariable-calculus vector-fields

share|cite|improve this question

asked Mar 24 at 16:34

UltradarkUltradark

3481518

$endgroup$

add a comment | 

$begingroup$

Here is a vector field:$$ vec F(x,y)=sin(x),sin(y), $$

where $x,y in (0,pi).$

How do you find the differential equation, that when solved gives the integral curves for this vector field? I made this plot on WolframAlpha:

ordinary-differential-equations multivariable-calculus vector-fields

share|cite|improve this question

asked Mar 24 at 16:34

UltradarkUltradark

3481518

$endgroup$

share|cite|improve this question

asked Mar 24 at 16:34

UltradarkUltradark

3481518

share|cite|improve this question

asked Mar 24 at 16:34

UltradarkUltradark

3481518

asked Mar 24 at 16:34

UltradarkUltradark

3481518

asked Mar 24 at 16:34

UltradarkUltradark

3481518

1 Answer
1

active

oldest

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0

$begingroup$

If we treat your slope field $F$ as a system of differential equations, we get the following:
beginalign*
fracdxdt &=F_x = sin(x) \
fracdydt &= F_y = sin(y)
endalign*

The trajectories for an autonomous system of differential equations can be calculated through the following formula:
$dy/dx =(dy/dt) / (dx/dt)$. Thus, the curves for you slope field will follow the following differential equation:
$$fracdydx = fracfracdydtfracdxdt = fracsin(y)sin(x)$$
This equation is separable and should be relatively easy to solve.

share|cite|improve this answer

edited Mar 24 at 20:07

answered Mar 24 at 17:07

HyperionHyperion

702111

$endgroup$

add a comment | 

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0

$begingroup$

If we treat your slope field $F$ as a system of differential equations, we get the following:
beginalign*
fracdxdt &=F_x = sin(x) \
fracdydt &= F_y = sin(y)
endalign*

The trajectories for an autonomous system of differential equations can be calculated through the following formula:
$dy/dx =(dy/dt) / (dx/dt)$. Thus, the curves for you slope field will follow the following differential equation:
$$fracdydx = fracfracdydtfracdxdt = fracsin(y)sin(x)$$
This equation is separable and should be relatively easy to solve.

share|cite|improve this answer

edited Mar 24 at 20:07

answered Mar 24 at 17:07

HyperionHyperion

702111

$endgroup$

add a comment | 

0

$begingroup$

If we treat your slope field $F$ as a system of differential equations, we get the following:
beginalign*
fracdxdt &=F_x = sin(x) \
fracdydt &= F_y = sin(y)
endalign*

The trajectories for an autonomous system of differential equations can be calculated through the following formula:
$dy/dx =(dy/dt) / (dx/dt)$. Thus, the curves for you slope field will follow the following differential equation:
$$fracdydx = fracfracdydtfracdxdt = fracsin(y)sin(x)$$
This equation is separable and should be relatively easy to solve.

share|cite|improve this answer

edited Mar 24 at 20:07

answered Mar 24 at 17:07

HyperionHyperion

702111

$endgroup$

add a comment | 

$begingroup$

If we treat your slope field $F$ as a system of differential equations, we get the following:
beginalign*
fracdxdt &=F_x = sin(x) \
fracdydt &= F_y = sin(y)
endalign*

The trajectories for an autonomous system of differential equations can be calculated through the following formula:
$dy/dx =(dy/dt) / (dx/dt)$. Thus, the curves for you slope field will follow the following differential equation:
$$fracdydx = fracfracdydtfracdxdt = fracsin(y)sin(x)$$
This equation is separable and should be relatively easy to solve.

share|cite|improve this answer

edited Mar 24 at 20:07

answered Mar 24 at 17:07

HyperionHyperion

702111

$endgroup$

share|cite|improve this answer

edited Mar 24 at 20:07

answered Mar 24 at 17:07

HyperionHyperion

702111

answered Mar 24 at 17:07

HyperionHyperion

702111

answered Mar 24 at 17:07

HyperionHyperion

702111