MHB Differential Equations: Direction Field

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To sketch a direction field for the differential equation y' = x - y + 1, one must map the sign of y' in the x-y plane. The equation y = x + 1 - m indicates that y' is negative when y > x + 1, zero when y = x + 1, and positive when y < x + 1. It is recommended to compute slopes at integer lattice points within a range, such as -2 to 2 for both x and y, to simplify the process. Using a calculator can help visualize the direction field, but it's advised to first practice sketching by hand for a better understanding. Ultimately, mastering the manual sketching of the direction field enhances comprehension of the solution curves.
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Sketch a direction field for the differential equation. Then use it to sketch three solution curves.
$y'=x-y+1$

I really need help drawing this, I'm super confused. :confused:
 
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ineedhelpnow said:
Sketch a direction field for the differential equation. Then use it to sketch three solution curves.
$y'=x-y+1$

I really need help drawing this, I'm super confused. :confused:

The so called 'direction field' is simply the mapping of the sign of y' in the x-y plane. So setting y' = m = const, You obtain...

y' = m -> y = x + 1 - m (1)

The (1) means that in all the points where y > x + 1 is y' < 0, in all the points where y = x + 1 is y' = 0 and in all the points where y < x + 1 is y' > 0...

Kind regards

$\chi$ $\sigma$
 
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what my book does is plug in values for x and y into the equation and then they compute a whole bunch of different slopes. Should I make my graph from like $-2\le x \le 2$ and $-2\le y \le 2$ ?
 
is there a way to do it on my calculator?
 
ineedhelpnow said:
what my book does is plug in values for x and y into the equation and then they compute a whole bunch of different slopes. Should I make my graph from like $-2\le x \le 2$ and $-2\le y \le 2$ ?

That's what I would do...and for simplicity only compute the slope at lattice points, that is, those points whose coordinates are integers. This will give you 25 points at which to compute a slope.

I imagine your calculator can draw direction fields...back when I was a student, we had to program our calculators to do this. :D
 
oh i think i got it
 
i had it but i mistakenly closed the document and i keep putting in the equation but it won't show up anymore. I am putting in the equation of the graph but it's also asking me for initial conditions.
 
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I really recommend you do this problem by hand first, and only then look at a computer generated plot of the field. You get much more of a feel for what's going on by actually getting in there and doing it yourself. :D
 

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