1. Separation of Variables Solving First Order Differential Equations

2. Solving ODEs What is Solving an ODE? Eliminating All Derivatives Explicit Form Implicit Form

3. This Chapter 1st Order (Only First Derivative) Linear and Nonlinear

4. Calculus Brain Teaser: ?

6. Today We will try to make problems look like:

7. Why? Want to “Get Rid of” This Derivative

8. Why? So we integrate the left side Have to integrate right side too

9. Separation of Variables No more derivatives! Implicit (General) Solution

10. Separation of Variables No more derivatives! Implicit (Specific) Solution If we havecan solve for C

11. Chain Rule Remember, y is a function of t

12. Chain Rule

14. So To Solve Think of it as: (Reversing the Chain Rule)

15. So To Solve Think of it as: Find by solving Keep equation balanced by solving

16. The whole process… For an equation of the form: (May need to manipulate equation to get here)

17. The whole process… For an equation of the form: Separate the variables

18. The whole process… For an equation of the form: Separate the variables is

19. The whole process… For an equation of the form: Separate the variables Integrate both sides Perhaps solve for y, or C (if initial condition)

20. A Simple Example

27. A Convenient Technique

30. “Cross Multiply”

31. A Convenient Technique

34. Integral Curves Is solved by: or Equation for an ellipse (for different values of C)

35. Integral Curves Plots of Solutions for Different Values of -C are called “Integral Curves” Integral Curves Show Different Behaviors for Different Initial Conditions

36. Integral Curves

39. In Summary To Solve an ODE, eliminate derivatives One method for first order linear/nonlinear ODES Separation of Variables (Reverse Chain Rule) Integral curves are solution curves for different values of C

40. Questions?