Slope Fields and Determing Behavior of Any Solution

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SUMMARY

The discussion centers on the differential equation y' = y² and the analysis of its slope field. It is established that the behavior of the solution y(t) as t approaches infinity is contingent upon the initial value y(0). Specifically, if y(0) > 0, then y(t) approaches infinity as t increases. Conversely, if y(0) < 0, y(t) approaches 0. The conversation also highlights the special case where y(0) = 0, indicating that y remains zero for all t.

PREREQUISITES
  • Understanding of differential equations, specifically first-order nonlinear equations.
  • Familiarity with slope fields and their graphical representation.
  • Knowledge of the behavior of solutions to differential equations as t approaches infinity.
  • Basic concepts of initial value problems in calculus.
NEXT STEPS
  • Explore the construction and interpretation of slope fields for various differential equations.
  • Study the stability of equilibrium solutions in differential equations.
  • Investigate the implications of initial conditions on the solutions of nonlinear differential equations.
  • Learn about special cases in differential equations, particularly the behavior at critical points like y = 0.
USEFUL FOR

Students and educators in mathematics, particularly those focusing on differential equations, as well as researchers interested in the qualitative analysis of solutions to nonlinear equations.

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Homework Statement


The differential equation is y' = y^2
Draw a direction field for the given differential equation.
Based on the direction field, determine the behavior of y as t →∞. If this behavior depends
on the initial value of y at t =0, describe this dependency.

Homework Equations


The Attempt at a Solution


Okay, so I plotted the slope field by evaluating the derivative at several different y-values. This is what I observed:

The solution curve certainly depends on the initial y-value when t=0. If particular solution y(t) to the DE has a solution of the form (t=0, y>0) (passes through a point), then as t----> infinity,
y(t)----> infinity. On the other hand, if some particular solution y(t) passes through a point of the form (t=0, y<0), then as t--->infinity, y(t)----> 0

Are these correct observations; and have I used terminology and notation correctly?

Also, when considering solutions that pass through the negative portion of the y-axis, will some approach y=0 more quickly than others?
 
Last edited:
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I am confused as to why you would say "On the other hand" for the initial value of y< 0 then say that its behavior is exactly the same as for y> 0.

And you seem to have left out an important special case- what if y= 0 for some t? And if y goes to infinity even if y is negative, how does it pass y= 0?
 
What you have pointed out, I believe I have fixed.

In the case that some solution passes through (0,0), then as t---> infinity, y will remain zero, right?
 

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