System of ODE's and direction field

Given the system of diffe-eq's:

[tex]x'(t)=x(t)+9y(t)[/tex]

[tex]y'(t)=-2x(t)-5y(t)[/tex]

I solved these ok. My question is, when graphing the solution curves on a direction field, I set up the direction field using the vector:

[tex](x+9y)\hat{i}+(-2x-5y)\hat{j}[/tex]

My question is, what is the relationship between this vector, and:

[tex]\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}[/tex]

Are they supposed to be equivalent?

Thanks!

 

Hey Tim,

Thanks for the response.

I meant to ask what is the relationship of the vector <x'(t),y'(t)> with dy/dx?

 

Hmm, it seems that dy/dx is the slope of the vector <x'(t),y'(t)>, is that the extent of the relationship between the two?

 

Ah ok. Thanks!