Section 1.1 Differential Equations & Mathematical Models

1. Section 1.1 Differential Equations & Mathematical Models

2. Differential Equations – “Equations with derivatives in them.” Examples: 1. 2. 3. y″ – 2 = 3xln(x) + y2

3. What is a solution to a differential equation?

4. A differential equation will often have infinitely many solutions. For example, here are some of the many solutions of 1. y = x2 2. y = x2 + e–x 3. y = x2 + 4e–x 4. y = x2 + 31429674e–x : : “Family of solutions”

5. Calculus Review: If then

6. A differential equation will usually have infinitely many solutions, but there times when a differential equation will have only one solution or no solutions. Example: (y′)2 + y2 = –1

7. Some differential equations will have solutions, but unfortunately we can't write them down (in terms of our elementary functions). Example:

8. General Solutions vs. Particular Solutions 1. Solve 2. Solve if y = 5 when x = 1. 3. Solve , y(1) = 5.

9. Initial Value Problem – consists of a differential equation along with an initial condition y(xo) = yo

10. Definition: Order The order of a differential equation is the order of the highest derivative appearing in it.

11. Expressing differential equations: Often we will be able to express 1st order differential equations as

12. Expressing differential equations: We will always be able to express. . . . 1st order differential equations in the form F(x, y, y′) = 0 2nd order differential equations in the form F(x, y, y′, y″) = 0 : nth order differential equations in the form F(x, y, y′, y″, y″′, . . . . , y(n)) = 0

13. Definition: Solution to a Differential Equation A function u(x) is a solution to the differential equation F(x, y, y′, y″, . . , y(n)) = 0 on an interval J if u, u′, u″, . . . , u(n) exist on J and F(x, u, u′, u″, . . . , u(n)) = 0 for all x on J.

14. Ex. 1 (a) Show that y(x) = 1/x is a solution to on the interval [1, 20].

15. Ex. 1 (b) Show that y(x) = 1/x is not a solution to on the interval [-20, 20].

16. Ex. 2 (a) Show that y1(x) = sin(x) is a solution to (y′ )2 + y2 = 1 (b) Show that y2(x) = cos(x) is a solution to (y′ )2 + y2 = 1

17. Partial Derivatives Ordinary Differential Equations vs. Partial Differential Equations

18. Section 1.2 Integrals as General & Particular Solutions

19. Ex. 1 Solve

20. Ex. 2 Solve

21. Ex. 3 Solve

24. Position - Velocity – Acceleration s(t) = position s′ (t) = velocity s″ (t) = acceleration Force = Mass x Acceleration

25. Ex. 5 A lunar lander is falling freely toward the surface of the moon at a speed of 450 m/s. Its retrorockets, when fired, provide a constant deceleration of 2.5 m/s2 (the gravitational acceleration produced by the moon is assumed to be included in the given deceleration). At what height above the lunar surface should the retro rockets be activated to ensure a "soft touchdown" (velocity = 0 at impact)?

26. Ex. 5 A lunar lander is falling freely toward the surface of the moon at a speed of 450 m/s. Its retrorockets, when fired, provide a constant deceleration of 2.5 m/s2 (the gravitational acceleration produced by the moon is assumed to be included in the given deceleration). At what height above the lunar surface should the retro rockets be activated to ensure a "soft touchdown" (velocity = 0 at impact)?

27. Section 1.3 Slope Fields & Solution Curves

28. Slope field for

29. Slope field for

30. Ex. 1 Sketch the slope field for y′ = –x

31. Ex. 2 Sketch the slope field for y′ = x2 + y2

32. Ex. 3 Examine some solution curves of On the following slope field, draw the solution curve which satisfies the initial condition of. . . . . (a)y(2) = –1 (b)y(–1) = 3 (c)y(0) = 0 (d)y(0) = 1

33. Calculus Review (definition of continuity): f (x) is continuous at xo if

34. Calculus Review (definition of continuity): f (x) is continuous at xo if f (x, y) is continuous at (xo, yo) if

35. Theorem I: Existence & Uniqueness of Solutions Suppose that f (x, y) is continuous on some rectangle in the xy-plane containing the point (xo, yo) in its interior and that the partial derivative fy is continuous on that rectangle. Then the initial value problem has a unique solution on some open interval Jo containing the point xo.

36. Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (a)

37. Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (b)

38. Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (c)

39. Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (d)

40. Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (e)

41. Ex. 4 Determine what this theorem says about the solutions in each of the following differential equations: (f)

42. Section 1.4 Separable Equations & Applications

43. Definition: Separable Differential Equation A first order differential equation is said to be separable if f (x, y) can be written as a product of a function of x and a function of y (i.e. ).

44. Definition: Separable Differential Equation A first order differential equation is said to be separable if f (x, y) can be written as a product of a function of x and a function of y (i.e. ). Examples: 1. 2. 3.

45. To solve a separable differentiable equation of the form we proceed as follows:

46. To solve a separable differentiable equation of the form we proceed as follows: h(y) dy = g(x) dx

47. To solve a separable differentiable equation of the form we proceed as follows: h(y) dy = g(x) dx (Then integrate both sides and solve for y, if this is possible.)

48. Ex. 1 Solve

49. Ex. 2 Solve

50. Justification for why this method for solving separable differentiable equations actually works.