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.
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).