1. Autonomous Differential Equations
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

2. Autonomous Differential Equations
If the independent variable does not appear in a D.E. it is called AUTONOMOUS. A few examples:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

3. Autonomous Differential Equations
If the independent variable does not appear in a D.E. it is called AUTONOMOUS. A few examples:
Notice that all of these equations are separable.
Also, There could be an equilibrium solution where y’=0. This is the first thing you should do when you see an autonomous D.E. – find any equilibrium solutions.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

4. Equilibrium solutions may occur when y’=0, so do that first:
We will solve all of these explicitly, but first we take a look at the equilibrium solutions and what they tell us.
Equilibrium solutions may occur when y’=0, so do that first:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

5. Equilibrium solutions may occur when y’=0, so do that first:
We will solve all of these explicitly, but first we take a look at the equilibrium solutions and what they tell us.
Equilibrium solutions may occur when y’=0, so do that first:
y(t)=0 is an equilibrium solution. Next we check on either side to determine whether it is stable or unstable.
For this we can use a PHASE LINE diagram. Plot the equilibrium points on a number line, then use the original D.E. to see what happens on either side.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

6. Equilibrium solutions may occur when y’=0, so do that first:
We will solve all of these explicitly, but first we take a look at the equilibrium solutions and what they tell us.
Equilibrium solutions may occur when y’=0, so do that first:
y(t)=0 is an equilibrium solution. Next we check on either side to determine whether it is stable or unstable.
For this we can use a PHASE LINE diagram. Plot the equilibrium points on a number line, then use the original D.E. to see what happens on either side.
For this case, when y<0, y’<0 and when y>0, y’>0 so we get the following phase line: y
Arrows pointing away from the equilibrium mean it is UNSTABLE.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

7. Here is the solution via separation of variables
Here is the solution via separation of variables. Notice that last step – we will see that trick all the time.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

8. Find equilibrium solutions by setting y’=0
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

9. Find equilibrium solutions by setting y’=0
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

10. Find equilibrium solutions by setting y’=0
Phase line diagram tells us the stability of each equilibrium. y 1
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

11. Find equilibrium solutions by setting y’=0
Phase line diagram tells us the stability of each equilibrium. y 1
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

12. Find equilibrium solutions by setting y’=0
Phase line diagram tells us the stability of each equilibrium. y 1
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

13. Find equilibrium solutions by setting y’=0
Phase line diagram tells us the stability of each equilibrium. y 1
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

14. Find equilibrium solutions by setting y’=0
Phase line diagram tells us the stability of each equilibrium. y 1
y=0 is stable
y=1 is unstable
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

15. y=0 is stable and has a basin of attraction (-∞,1)
-the set of initial values that end up at a particular equilibrium point
y=0 is stable and has a basin of attraction (-∞,1)
y=1 is unstable and has basin of attraction y=1 (only a single value)
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

16. Find equilibrium solutions by setting y’=0
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

17. Find equilibrium solutions by setting y’=0
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

18. Find equilibrium solutions by setting y’=0
Phase line diagram tells us the stability of each equilibrium. y -1 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

19. Find equilibrium solutions by setting y’=0
Phase line diagram tells us the stability of each equilibrium. y -1 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

20. Find equilibrium solutions by setting y’=0
Phase line diagram tells us the stability of each equilibrium. y -1 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

21. Find equilibrium solutions by setting y’=0
Phase line diagram tells us the stability of each equilibrium. y -1 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

22. Find equilibrium solutions by setting y’=0
Phase line diagram tells us the stability of each equilibrium. y -1 2
y=-1 is stable
y=2 is unstable
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

23. y=-1 is stable and has a basin of attraction (-∞,2)
-the set of initial values that end up at a particular equilibrium point
y=-1 is stable and has a basin of attraction (-∞,2)
y=2 is unstable and has basin of attraction y=2 (only a single value)
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

24. Here is the slope field for .
Notice the equilibrium solutions at y=0,±∏,… where the slope is horizontal. We can find these solutions by setting y’=0.
All of these equilibria are unstable, which is easy to see from the diagram.
Also, at y=±∏/2,… there are points where the slope is vertical, where y’ is undefined.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB