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# Show that the isoclines of y′ = x + y form a one-parameter family of lines

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

Show that the isoclines of the differential equation

form a one-parameter family of straight lines. Make a plot of the isoclines corresponding to the slopes . Using these isoclines, construct a direction field and sketch the integral curve passing through the origin. Identify one of the integral curves is also an isocline.

The isoclines of are the curves which implies . These are straight lines with slope . The integral curve passing through the origin is . The isocline is also an integral curve.

### One comment

1. Mike says:

The integral curve passing through the origin is not y=-x. In that case y’=-1, which is not x+y=0 (as it should be if it is the solution of y’=x+y).