What differential equation corresponds to this vector field?

Question

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:

Answer 1

If we treat your slope field $F$ as a system of differential equations, we get the following: \begin{align*} \frac{dx}{dt} &=F_x = \sin(x) \\ \frac{dy}{dt} &= F_y = \sin(y) \end{align*}

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: $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\sin(y)}{\sin(x)}$$ This equation is separable and should be relatively easy to solve.