Solve first order ordinary differential equation (ODE).

Question

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