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Plot the isoclines of the equation y = xy′ + y′′

If y is the solution of a differential equation, the points at which y' has a constant value C lie on a line for each C. This line is called an isocline.

Plot some isoclines and construct a direction field of the equation

    \[ y = x \frac{dy}{dx} + \left( \frac{dy}{dx} \right)^2. \]

Determine a one-parameter family of solutions of this equation from the appearance of the direction field.


One comment

  1. Mohammad Azad says:

    Let the slope be fixed say y’=c then y=xy’+y’^2=cx+c^2 Draw a bunch of these lines carefully, these lines are isoclines, and you use them to plot the direction field by picking points on these lines and drawing short segments with slope y’ at that point, for example we let c=1 and draw y=x+1 and on this line we draw short segments of slope c, in this case the short segments will line on the isocline y=x+1 and this makes it easy, when you draw enough of them you might notice an envelope.

    You can also use some apps to draw these fields like 2D ODE GRAPHER https://play.google.com/store/apps/details?id=angus.planarodenumerics

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