I'm struggling a bit in finding the integralcurve of a given tangential field. Let's say we have (easy example, I think) the following tangential field: $X=(x+y) \delta_x + (y-x) \delta_y$.
How do I have to proceed to get the integral curve?
I know that I have to find an equation like here: (Example 1.b). But how should the equation look like for my example? (And why?) I think this information should be enough for me to solve the task.
Thanks a lot for the help!
Following the notations in the note, let $\gamma(t)=(x(t),y(t))$ be the integral curve for $X$. Then the equation for the integral curve would become $x'(t)=x(t)+y(t), y'(t)=y(t)-y(t)$ in this case. (It seems a typo here?) Then you can find the solution for this ODE as this is a system of linear ODE. (More accurately, you can claim that there exists a unique solution given an initial condition, but finding an explicit form of the solution may be hard in general.)
Edit: After finding the system of ODEs for the integral curve, all you need to consult with is some techniques in standard ODE course; for example, in the current example, the ODE is 1st order homogeneous and linear, so you can rewrite the given ODE as
$\mathbf{x}'=A\mathbf{x}$, where $\mathbf{x}=(x(t),y(t))^t$ and $A=\begin{pmatrix} 1 &1 \\ -1 & 1 \end{pmatrix}$.
And then the eigenvalues of $A$ are $1\pm i$ whose corresponding eigenvectors are $(1 ,\pm i)^t$ respectively. Thus the general solution of this ODE is $\mathbf{x}=ae^{(1+i)t}(1,i)^t+be^{(1-i)t}(1,-i)^t$ for constants $a,b$ that satisfy the initial condition(say, a point where your integral curve starts at).