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C .3.1 - Differential Equations

C .3.1 - Differential Equations. Calculus - Santowski. Lesson Objectives. 1. Become familiar with a definition of and terminology involved with differential equations 2. Solve differential equations with and without initial conditions

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C .3.1 - Differential Equations

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  1. C.3.1 - Differential Equations Calculus - Santowski Calculus - Santowski

  2. Lesson Objectives • 1. Become familiar with a definition of and terminology involved with differential equations • 2. Solve differential equations with and without initial conditions • 2. Apply differential equations in a variety of real world applications Calculus - Santowski

  3. (A) Definitions • A differential equation is any equation which contains derivatives • Ex: • Ex: Newton’s second law of motion can also be rewritten as a differential equation: • F = ma which we can rewrite as Calculus - Santowski

  4. (A) Definitions • The order of a differential equation refers to the largest derivative present in the DE • A solution to a differential equation on a given interval is any function that satisfies the differential equation in the given interval Calculus - Santowski

  5. (B) Example • Show that is a solution to the second order DE for x > 0 • Why is the restriction x > 0 necessary? • Are the equations listed below also solutions?? Calculus - Santowski

  6. (C) Initial Conditions • As we saw in the last example, a DE will have multiple solutions (in terms of many functions that satisfy the original DE) • So to be able to identify or to solve for a specific solution, we can given a condition (or a set of conditions) that will allow us to identify the one single function that satisfies the DE Calculus - Santowski

  7. (D) Example - IVP • Show that is a solution to the second order DE given the initial conditions of • What about the equation Calculus - Santowski

  8. (D) Examples • Ex 1. Show that is a general solution to the DE • Ex 2. What is the solution to the IVP Calculus - Santowski

  9. (E) Further Examples • Ex 1. What is the solution to the first order differential equation (FODE) • Sketch your solution on a graph. • Then solve the IVP given the initial condition of (-1,1) Calculus - Santowski

  10. (E) Further Examples • Ex 2. Solve the IVP • How would you verify your solution? Calculus - Santowski

  11. (E) Further Examples • Ex 3. What is the solution to the first order differential equation (FODE) • Then solve the IVP given the initial condition of Calculus - Santowski

  12. (E) Further Examples • Ex 4. What is the solution to the first order differential equation (FODE) • Then solve the IVP given the initial condition of Calculus - Santowski

  13. (E) Further Examples • Ex 5. What is the solution to the first order differential equation (FODE) • (Note that this time our solution simply has an integral in the solution – FTC) Calculus - Santowski

  14. (F) Motion Examples • If a body moves from rest so that its velocity is proportional to the time elapsed, show that v2 = 2as where s = displacement, t is time and v is velocity and a is a constant (representing ….?) Calculus - Santowski

  15. (F) Motion Examples • A particle is accelerated in a line so that its velocity in m/s is equal to the square root of the time elapsed, measured in seconds. How far does the particle travel in the first hour? Calculus - Santowski

  16. (F) Motion Examples • A stone is tossed upward with a velocity of 8 m/s from the edge of a cliff 63 m high. How long will it take for the stone to hit the ground at the foot of the cliff? Calculus - Santowski

  17. (G) In Class Worksheets • We will work through selected questions from handouts from several other Calculus textbooks • (a) Finney et al, §6.1, p327, Q1-24 • (b) Dunkley et al, §7.3, p356, Q1-3 Calculus - Santowski

  18. (H) DE on TI-89 Calculus - Santowski

  19. (H) DE on TI-89 • So the last slides have shown how to solve DE on the TI-89 as well as preparing graphs (which we will explore in more detail in a future lesson) Calculus - Santowski

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