# -b.1.1.4 directional field as t \to \infny

• MHB
• karush
In summary, the conversation discusses solving the differential equation $y'=-1-2y$ and provides step-by-step instructions for finding the solution. It is shown that the solution is of the form $y = -\frac{1}{2} + Ce^{-2t}$ and as $t$ goes to infinity, the value of $y$ approaches $\frac{1}{2}$. The conversation also mentions using Desmos to visualize the solution.
karush
Gold Member
MHB
from the spoiler its #4 hope I got instructions right

$y'=-1-2y$
$\begin{array}{ll} rewrite &y'+2y=-1\\ exp &u(t)=\exp\ds\int 2 \, dx=e^{2t}\\ product &(e^{2t}y)'=-e^{2t}\\ &e^{2t}y=\ds\int -e^{2t} \, dt=\dfrac{-e^(2 t)}{2}+c\\ hence &y(t)=\dfrac{-e^{2t}}{2} + \dfrac{c}{e^{2t}}\\ t \to \infty &=-\dfrac{1}{2}+0\\ so &y \to \dfrac{1}{2}\textit{ as t} \to \infty \end{array}$

ok this is an early problem but probably some comments to make it more understandable
hopefully correct
I was going to try tikz on this but probably 3 pages of code so its a desmos templatehttps://dl.orangedox.com/geAQogxCM0ZYQLaUju

I get

$y = -\dfrac{1}{2} + Ce^{-2t}$

From $y'= -1- 2y$, you don't have to write it as $y'+ y= -1$.
You can rewrite it as $\frac{dy}{dx}= -1-2y$ and then separate as $\frac{dy}{2y+ 1}= -dx$..

Now integrate both sides- on the left, let $u= 2y+ 1$ so that $du= 2dy$ and $dy= \frac{1}{2}du$
$\frac{1}{2}\int \frac{du}{u}=\frac{1}{2}ln(u)+ c_1= \frac{1}{2} ln(2y+ 1)= ln(\sqrt{2y+ 1})$ and $-\int dx= -x+ c_2$.

Combining the two contants into "c", $ln(\sqrt{2y+ 1})= -x+c$. Taking the exponential of both sides $\sqrt{2y+1}= e^{-x+ c}= Ce^{-x}$ where $C= e^c$.

Square both sides: $2y+ 1= \left(Ce^{-x}\right)^2= C'e^{-2x}$ where $C'= C^2$.
$2y= C'e^{-2x}- 1$
$y= C''e^{-2x}- \frac{1}{2}$ where $C''= C'/2$

And that goes to $-\frac{1}{2}$ as x goes to infinity.

Hey, wasn't that fun!

## 1. What is a -b.1.1.4 directional field as t \to \infny?

A -b.1.1.4 directional field as t \to \infny is a mathematical concept used in the study of differential equations. It represents the behavior of a system as time approaches infinity, and is often used to make predictions about the long-term behavior of a system.

## 2. How is a -b.1.1.4 directional field as t \to \infny calculated?

The calculation of a -b.1.1.4 directional field as t \to \infny involves solving the differential equation and finding the limit of the solution as t approaches infinity. This can be done analytically or numerically using various mathematical techniques.

## 3. What does a -b.1.1.4 directional field as t \to \infny tell us about a system?

A -b.1.1.4 directional field as t \to \infny provides information about the long-term behavior of a system. It can tell us if the system will approach a stable equilibrium, oscillate between different states, or exhibit chaotic behavior. This information is useful in understanding the overall behavior of a system.

## 4. How is a -b.1.1.4 directional field as t \to \infny used in real-world applications?

-b.1.1.4 directional field as t \to \infny is used in various fields of science and engineering to model and predict the behavior of complex systems. It is commonly used in physics, biology, economics, and other disciplines to understand and analyze real-world phenomena.

## 5. Are there any limitations to using a -b.1.1.4 directional field as t \to \infny?

Like any mathematical model, there are limitations to using a -b.1.1.4 directional field as t \to \infny. It assumes that the system is stable and does not take into account external factors that may affect the system's behavior. It is also based on simplifying assumptions and may not accurately reflect the complexity of a real-world system.

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