Understanding the Direction Field of y'=cos(πx)

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SUMMARY

The discussion focuses on solving the differential equation y' = cos(πx), resulting in the general solution y = sin(πx)/π + C. The user expresses confusion regarding the plotting of the direction field due to the nature of the cosine function, which restricts y' values to the range of [-1, 1]. The user attempts to plot points for y' = 0, 1, and -1 but struggles with the concept of isoclines, noting that they appear as discrete points rather than continuous curves.

PREREQUISITES
  • Understanding of differential equations, specifically first-order equations.
  • Familiarity with trigonometric functions, particularly the cosine function.
  • Knowledge of direction fields and their significance in visualizing solutions to differential equations.
  • Basic graphing skills to plot functions and interpret isoclines.
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  • Study the concept of direction fields in differential equations using tools like GeoGebra.
  • Learn about isoclines and their role in sketching solution curves for differential equations.
  • Explore the implications of periodic functions, such as cosine, on the behavior of solutions.
  • Practice plotting direction fields for various differential equations to gain confidence.
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Students and educators in mathematics, particularly those studying differential equations, as well as anyone interested in visualizing solutions through direction fields.

ajayguhan
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This isn't a home work, so only I'm posting here

When we solve y'=cos(πx) we'll get y=sin(πx)/π +C. But plotting direction field seems little confusing for me

Since y' is a function of cosine , the value of y' must be within [-1,1].

If i took y' as 0 and tried to plot it and I'm getting set of points such as (...-2.5, -1.5, -0.5, 0.5, 1.5, 2.5,...)

I can't draw a isocline since here it is discontinuois.

Similarly by setting y' has 1, -1 we'll get points such as (...-4, -2, 0, 2, 4...) & (...-5, -3, -1, 1, 3, 5...)

Now how the solution curve should be plotted from the direction field, without isocline (i mean here isocline is just set of points not a curve)...
 
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