1. Calculus II (MAT 146) Dr. Day Monday, March 19, 2018
Differential Equations (Chapter 9)
What is a Differential Equation? (9.1)
What is a Solution to a Differential Equation? (9.1)
Graphical Representations of Solutions to Differential Equations (9.2)
Monday, March 19, 2018
MAT 146

2. What is a Differential Equation?
A differential equation is an equation that contains one or more derivatives.
Here’s a differential equation you have already solved:
y’ = 2x
What is the solution of this differential equation?
Monday, March 19, 2018
MAT 146

3. What is a Solution to a Differential Equation?
A general solution to a differential equation is a family of functions that satisfies a given differential equation.
A particular solution to a differential equation (also called the solution to an initial-value problem) is a particular function that satisfies both a given differential equation and some specified ordered pair for the function.
Monday, March 19, 2018
MAT 146

4. DE Warm-Ups
For the differential equation here, what are the constant solutions?
Solve this initial-value problem:
Monday, March 19, 2018
MAT 146

5. Monday, March 19, 2018
MAT 146

6. Monday, March 19, 2018
MAT 146

7. Monday, March 19, 2018
MAT 146

8. Monday, March 19, 2018
MAT 146

9. Monday, March 19, 2018
MAT 146

10. Monday, March 19, 2018
MAT 146

11. Monday, March 19, 2018
MAT 146

12. knowing that (0,0) satisfies y
Solve the differential equation graphically by generating a slope field and then sketching in a solution:
y’ = 2x – y
knowing that (0,0) satisfies y
Monday, March 19, 2018
MAT 146

13. with initial conditions (0,0)
y’ = 2x – y
with initial conditions (0,0)
Monday, March 19, 2018
MAT 146

14. Monday, March 19, 2018
MAT 146

15. Monday, March 19, 2018
MAT 146

16. Monday, March 19, 2018
MAT 146

17. Monday, March 19, 2018
MAT 146

18. Monday, March 19, 2018
MAT 146

19. Monday, March 19, 2018
MAT 146

20. Monday, March 19, 2018
MAT 146

21. Solving Differential Equations
Solve for y: y’ = −y2
Monday, March 19, 2018
MAT 146

22. Separable Differential Equations
Monday, March 19, 2018
MAT 146

23. Separable Differential Equations
Solve for y: y’ = 3xy
Monday, March 19, 2018
MAT 146

24. Separable Differential Equations
Solve for z: dz/dx+ 5ex+z = 0
Monday, March 19, 2018
MAT 146

25. Separable Differential Equations
Monday, March 19, 2018
MAT 146

26. Separable Differential Equations
Monday, March 19, 2018
MAT 146