Differential Equations: Direction Field

Click For Summary

Discussion Overview

The discussion revolves around sketching a direction field for the differential equation $y' = x - y + 1$ and using it to draw solution curves. Participants express confusion about the process and seek guidance on how to approach the task, including the use of calculators and manual methods.

Discussion Character

  • Exploratory
  • Homework-related
  • Technical explanation

Main Points Raised

  • Some participants suggest that the direction field represents the sign of $y'$ in the x-y plane, leading to the equation $y = x + 1 - m$ for constant slopes.
  • One participant proposes computing slopes at integer lattice points within a specified range of $-2 \leq x \leq 2$ and $-2 \leq y \leq 2$ to simplify the process.
  • Another participant inquires about the possibility of using a calculator to generate the direction field.
  • There is a suggestion to first sketch the direction field by hand to gain a better understanding before using computer-generated plots.
  • One participant expresses frustration about technical issues with their document and the need for initial conditions when inputting the equation into their graphing tool.

Areas of Agreement / Disagreement

Participants generally express confusion and seek clarification on the process of sketching the direction field. There are multiple approaches suggested, including manual sketching and using calculators, but no consensus on the best method is reached.

Contextual Notes

Participants mention varying methods for computing slopes and the potential need for initial conditions, indicating that assumptions about the approach may differ. The discussion reflects uncertainty about the best practices for sketching direction fields.

Who May Find This Useful

This discussion may be useful for students learning about differential equations, particularly those interested in visualizing direction fields and solution curves, as well as those seeking different methods for approaching such problems.

ineedhelpnow
Messages
649
Reaction score
0
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:
 
Physics news on Phys.org
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

$\chi$ $\sigma$
 
Last edited:
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.
 
Last edited:
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
 

Similar threads

Undergrad Direction Fields and Isoclines
  • · Replies 1 ·
Replies
1
Views
800
Undergrad Why Lagrange’s method of solving Pp + Qq=R works?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Drawing Direction Fields for Higher Order ODEs
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Partial Differential Equation solved using Products
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Solving Differential Equation Using Reduction of Order
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad General solution vs particular solution
  • · Replies 52 ·
2
Replies
52
Views
8K
Graduate Integrability of Systems of Differential Equations
  • · Replies 5 ·
Replies
5
Views
3K
Insights Differential Equation Systems and Nature
  • · Replies 14 ·
Replies
14
Views
5K
Undergrad On sub and super solutions: Teschl and others
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Integral-form change of variable in differential equation
  • · Replies 1 ·
Replies
1
Views
2K