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**MAT244--Lectures & Home Assignments / Describing Direction Fields**

« **on:**September 20, 2018, 04:26:38 AM »

To what extent are we expected to describe direction fields?

For example, consider the differential equation from the homework question:

$y' = e^{-t} + y$

I used the online ODE plotter to determine its behaviour.

As $t → ∞$, $y$ diverges from $0$.

$y$ approaches $∞$ when $t > 0$, and $y$ approaches $-∞$ when $t < 0$.

At $t = 0$ and $y = -1$, the value for $y'$ is zero and the slope line is horizontal.

The question asks us to determine the behaviour of $y$ as $t → ∞$, and whether this behaviour depends on the initial value of $y$ at $t = 0$. Is the explanation above enough?

For example, consider the differential equation from the homework question:

**Section 1.1 #28**$y' = e^{-t} + y$

I used the online ODE plotter to determine its behaviour.

As $t → ∞$, $y$ diverges from $0$.

$y$ approaches $∞$ when $t > 0$, and $y$ approaches $-∞$ when $t < 0$.

At $t = 0$ and $y = -1$, the value for $y'$ is zero and the slope line is horizontal.

The question asks us to determine the behaviour of $y$ as $t → ∞$, and whether this behaviour depends on the initial value of $y$ at $t = 0$. Is the explanation above enough?