How do you find isoclines?

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I can find many websites that show a slope field, an answer, and the isoclines but for the life of me I cannot figure out the relationship between slope fields and isoclines!!!

I need a nudge in the right direction please!
 

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When you have a system of ode's, your solution will be a parametrized curve in the plane (space), i.e. [itex]\{x(t),y(t)\}[/itex]. If you derivate such curve, you obtain a vector tangent to such curve given by [itex]T=\{\dot{x}(t),\dot{y}(t)\}[/itex], where the dot denotes derivations with respect to time. From your calc & geometry classes, you should remember that the slope of the tangent vector is given by (using the chain rule):

[tex]m=\frac{d y/dt}{dx/dt}=\frac{dy}{dx}[/tex].

And there you go. If you have a given system

[tex]\begin{array}{l} \dot{x}(t)=f(x,y,t) \\ \dot{y}(t)=g(x,y,t)\end{array}[/tex]

then the isoclines will be the curves where the slope

[tex]m=\frac{g(x,y,t)}{f(x,y,t)}[/tex]

remains constant. Of particular importance are the nullclines ([itex]m=0[/itex] and [itex]m=\infty[/itex]). (why?)
 
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