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I have a parametrized curve $$f(t)=(x(t),y(t))$$ such that $f(t)=x(t)+y(t)$ and the tangent vector $X(f(t))=(-p(t),q(t))$

then ,the curve satisfy the first ordinary differential equation is get by equation such that

$$\langle \dot{f}(t),X(f(t))\rangle=-\frac{\partial f(t)}{\partial x(t)}p(t)+\frac{\partial f(t)}{\partial y(t)}q(t)=0$$

I need to solve this equations and plot it(Or only plot it as a special case for geometric properties

Can they be considered as an first order ordinary differential equation (ODE).

Is there a solution? Is it possible to use a software program for solving and drawing

Thanks for the help

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    $\begingroup$ What is the meaning of those letters, which functions are given, what are the unknowns? The only explicitely given entity is $0$, and you won't get much of a drawing out of that, I'm afraid. So I think your statement of the problem is somewhat incomplete. $\endgroup$ – Professor Vector Jun 13 '17 at 6:34
  • $\begingroup$ Equation $(1)$ actually follows from equation $(2)$. $\endgroup$ – John Wayland Bales Jun 13 '17 at 6:34
  • $\begingroup$ @John Wayland Bales How exactly would if follow? I'm especially interested in the magical appearance and meaning of $g$. $\endgroup$ – Professor Vector Jun 13 '17 at 6:47
  • $\begingroup$ It's too long for a comment, I'm working on it and will submit it as a partial answer. $\endgroup$ – John Wayland Bales Jun 13 '17 at 6:48
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    $\begingroup$ It would be advisable to write every bit of information, assumptions, definitions,... in your question. I'm pretty sure you can't write DSolve[pde, f[x, y], {x, y}] in Mathematica without definining f, first. And there's a "pde" in that line, and "ODE" in the title. So what is your question? $\endgroup$ – Professor Vector Jun 13 '17 at 6:53

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