306 CHAPTER 5 Series and Differential Equations
From the above slope field it appears that there might be alinear
solution of the formy =mx+b. We can check this by substituting
into the differential equation:
m=d
dx
(mx+b) = mx+b−x,which immediately implies thatm= 1 andb=m= 1.
Example 2. The logistic differential equation y′ = 3y(1−y) has
slope field given to the right. What we should be able to see from this
picture is that if we have an initial condition of the formy(0) =y 0 >0,
then the solutiony=y(x) will satisfy limx→∞y= 1.
Exercises
- For each of the slope fields given below, sketch a solution curve
y=y(x) passing through the initial point (x 0 ,y 0 ).