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I need a nudge in the right direction please!

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- Thread starter ssb
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In summary, slope fields and isoclines are related because they both involve the derivatives of a parametrized curve in the plane. The slope of the tangent vector of the curve is given by the chain rule, and the isoclines are curves where the slope remains constant. Nullclines, where the slope is either 0 or infinity, are particularly important in this relationship.

- #1

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I need a nudge in the right direction please!

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- #2

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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?)

[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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Isoclines are lines on a slope field that have the same slope at every point. In other words, they represent the points where the derivative (or slope) of a function is constant. To find isoclines, you can use the slope field to identify points where the slope remains the same. These points will form a line, which is the isocline. In some cases, you may need to plot multiple points and connect them to get a better understanding of the isocline. Additionally, you can use the equation of the function to determine the isoclines, as the derivative of a function is equal to the slope of the tangent line at any given point. I recommend practicing with different slope fields and functions to get a better understanding of the relationship between them. Hope this helps!

An isocline is a mathematical term used in dynamical systems to describe a curve or line where a particular quantity remains constant. It is often used to illustrate the equilibrium or steady-state solutions of a system.

Finding isoclines helps us understand the behavior and stability of a dynamical system. By identifying the isoclines, we can determine the equilibrium points and analyze the system's behavior over time.

To graph an isocline, you need to plot the points where the quantity remains constant and connect them to form a curve or line. This can be done by setting the quantity to a specific value and solving for the other variables in the system.

Yes, isoclines can intersect at different points on the graph. These points represent the equilibrium solutions of the system, where the quantities remain constant.

Isoclines can be used to determine the stability of a system. If the isoclines intersect at a single point, the system is stable and will reach a steady-state. If the isoclines form closed loops, the system is unstable and will not reach equilibrium. Additionally, the slope of the isoclines can indicate the direction in which the system will move over time.

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