I have been stuck on this question for very long now
Given the vector field $$ F(x, y) = (5x + 2y, 3x + 7y) $$ And the ODE system: $$ x' = 6x + 9y\\ y' = 1x + 4y. $$ Determine the integral curves to $F$ by finding the general solution to the ODE system.
My problem:
I have tried to solve the general solution and got: $$ v(t) = c(2,-5)e^{-t} + d(5, 2)e^{5t}. $$ But I don't know how to get the integral curves.
It seems that you've arrived at the solutions $$x = ce^{-t} + de^{5t}, \; y = -ce^{-t} + de^{5t}.$$ From here, note that $x+y = 2de^{5t}$ and $x-y = 2ce^{-t}$. For any given solution, $c$ and $d$ are fixed. So we hence have that $$(x+y)(x-y)^5 = 2d(2c)^5 e^{5t} e^{-5t} = 64c^5d$$ which is constant. So we have integral curves of form $$(x+y)(x-y)^5 = a,$$ for $a \in \mathbb R$.