Differential Equation dy/dx= 1/y: describe properties of this equation and their effect on the slope field that represents the family of solutions to this DE

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- Dec 9th 2012, 07:13 PMcamjensonSlope Fields
Differential Equation dy/dx= 1/y: describe properties of this equation and their effect on the slope field that represents the family of solutions to this DE

- Dec 9th 2012, 08:32 PMMarkFLRe: Slope Fields
What do the isoclines for this direction field look like?

- Dec 9th 2012, 08:46 PMcamjensonRe: Slope Fields
i don't know that word ha im sorry

- Dec 9th 2012, 09:05 PMMarkFLRe: Slope Fields
An isocline is the locus of points in the plane such that the slope is always the same. For example, where is the slope always 1?

- Dec 9th 2012, 09:11 PMcamjensonRe: Slope Fields
when y=1?

- Dec 9th 2012, 09:20 PMMarkFLRe: Slope Fields
Exactly! How about where the slope is always k, where k is an arbitrary constant?

- Dec 9th 2012, 10:04 PMcamjensonRe: Slope Fields
when y=0?

- Dec 9th 2012, 10:14 PMMarkFLRe: Slope Fields
No. How did you find the slope is always 1 along the line y = 1?

- Dec 9th 2012, 10:16 PMcamjensonRe: Slope Fields
because 1/1 is 1

- Dec 9th 2012, 10:24 PMMarkFLRe: Slope Fields
Right. We could state:

$\displaystyle \frac{dy}{dx}=\frac{1}{y}=1\,\therefore\,y=1$

So, if we generalize, we can state:

$\displaystyle \frac{dy}{dx}=\frac{1}{y}=k\,\therefore\,y=\frac{1 }{k}$

The whole point to this is that we find the isoclines are horizontal lines, and to find the isocline where the slope is $\displaystyle k$, we use the line $\displaystyle y=\frac{1}{k}$. Finding isoclines is a method to make sketching the direction field easier.

What kind of symmetry do we observe in the direction field? - Dec 9th 2012, 10:42 PMcamjensonRe: Slope Fields
the higher y is, the smaller the slope

- Dec 9th 2012, 10:57 PMMarkFLRe: Slope Fields
Yes, that's true for positive y. For negative y, the opposite is true. The further we get from the x-axis, the smaller the magnitude of the slope is.

The symmetry I am speaking of comes from $\displaystyle y'(-y)=-y'(y)$. So, what kind of symmetry can we expect to find in the solution space? - Dec 9th 2012, 11:03 PMcamjensonRe: Slope Fields
isn't it the same thing? or are you supposed to take the antiderivative of 1/y?

- Dec 9th 2012, 11:29 PMMarkFLRe: Slope Fields
You can solve the ODE by integration, but for this exercise you are expected to merely look at the nature of the solution from the direction field.

By looking at the direction field, we can see that the solutions will be symmetric across (or reflected about) the x-axis. - Dec 10th 2012, 12:15 AMcamjensonRe: Slope Fields
how can we draw a solution curve through (-1,0) if that point doesnt exist?