MHB How Does the Initial Condition Affect the Behavior of Y in Nonlinear First ODEs?

[cbarker1]

Hello,

I need some help with describing the dependency with some nonlinear First ODEs.
The question: "Using DField, [a java program to graph first-ODEs], draw the direction field for the DE. Based on the direction field, e the determine behavior of y as t approaches infinity. If this behavior depends on the initial conditions, describes this dependency?

1. $y'=t*e^{-2t}-2y$

The behavior of y as t approaches infinity is approaching zero.https://www.physicsforums.com/attachments/4773

 

[Ackbach]

I'm a tad confused by what you're asking. The DE you've provided is a linear, first-order ODE. Using standard methods, you can arrive at the exact solution
$$y(t)=\frac{x^2+C}{2 e^{2t}}.$$
Is the general behavior dependent on the initial condition? I should say not. As $t\to\infty$, the solutions are going to zero, regardless of where you start (assuming the initial condition has $t\ge 0$.)

Does this answer your question?

 

[cbarker1]

The instructor wants me to read the slope fields to determine what the behavior of the solution.

 

[Ackbach]

So, if you do that, you notice that all the field lines are going towards $y=0$, provided that $t>0$. Do you see that? So then, the only real difference in behavior occurs if the initial $y$ position is negative or positive. That determines whether the solution will increase or decrease. This is all information you can glean from the slope field.