1 Chapter 8 Ordinary differential equation Mathematical methods in the physical sciences 3rd edition Mary L. Boas Lecture 5 Introduction of ODE

2 1. Introduction (differential equation) - A great many applied problems involve rates, that is, derivatives. An equation containing derivatives is called a differential equation. - If it contains partial derivatives, it is called a partial differential equation; otherwise it is called an ordinary differential equation. ex 1) Newton’s equation ex 2) Heat transfer ex 3) RLC circuit

3 - order of a differential equation : order of the highest derivative in the equation - (non)Linear differential equation

Note 1 : A solution of a differential equation (in the variable x and y) is a relation between x and y which, if substituted into the differential equation, gives an identity.  If you come up with a function to give an identity, that should be a solution of the differential equation. Example 1) Example 2) In order to verify if your solutions are correct, put the solutions into the equations and check the identity. 4

5 Example 3)Find the distance which an object falls under gravity in t seconds if it starts from rest. Note 2 - First order DE  one arbitrary integration constant (IC) - Second order DE  two ICs - N-th order DE  # of ICs is n - General solution with arbitrary IC - Particular solution determined by the boundary condition or initial condition

6 Example 4)Find the solution which passes through the origin and (ln2, 3/4)

7 2. Separable equations - Separable equation ex) y terms in one side and x terms in the other side  the equation is separable. Example 1) Radioactive substance decay rate

8 Example 2)

9 - Orthogonal trajectories: ex) lines of force intersect the equipotential curves at right angles.

10 Chapter 8 Ordinary differential equation Mathematical methods in the physical sciences 3rd edition Mary L. Boas Lecture 6 First order ODE

11 3. Linear first-order equations - Linear first-order equation

12

13 Example 1)

14 Example 2) Radium decays to radon which decays to polonium. If at t=0, a sample is pure radium, how much radon does it contain at time t (created and simultaneously decay)? N_0 = # of radium atoms at t=0 N_1= # of radium at time t N_2 = # of radon atoms at time t, lambda_1, lambda_2 = decay constants for Ra and Rn.

15

16 4. Other methods for first-order equations 1) Bernoulli equation It is not linear, but, is easily reduced to a linear equation by making the change of the variable.

17 2) Exact equations; integrating factors

18 ex. 1) ex. 2)

19 3) Homogeneous equations - The above equation can be reduced to a separable equation in variable v=y/x and x.

20 Prob. 8)

21 4) Change of variables : If a differential equation contains some combination of the variable x, y, we try replacing this combination by a new variable. cf. Problem 11.

22 Chapter 8 Ordinary differential equation Mathematical methods in the physical sciences 3rd edition Mary L. Boas Lecture 7 Second order ODE I

23 5. Second order linear equations with constant coefficients and zero right-hand side Example 1. 1) Auxiliary equation

24 Comment. Here, we can see that solving the second-order linear differential equation (y’’+my’+ny=0) is quite similar to solving the second order normal equation (D 2 +mD+n=0). We know that there are three cases for the solutions of the second order equation, two real numbers, single real number, and two complex numbers. The first case of DE corresponds to the equation with the two real solutions. How about the others cases?

25 2) Equal roots of the auxiliary equation

26 3) Complex conjugate roots of the auxiliary equations - The roots of auxiliary equations are complex. Example 2)

27 Example 3) motion of a mass oscillating at the end of a spring ‘We can determine two unknown constants using initial conditions.’ Example 4. Initial condition: at t=0, y=-10, y’=0

28 Example 5. Considering the friction,

29

30 - Underdamped oscillator

31 - Critically/over-damped

32 REVIEW

33 Chapter 8 Ordinary differential equation Mathematical methods in the physical sciences 3rd edition Mary L. Boas Lecture 8 Second order ODE II

34 6. Second-order linear equations with constant coefficient and right- hand side not zero “The general solution of an inhomogeneous DE is the combination of y_c and y_p” 1) solution of an inhomogeneous DE

35 2) Inspection of particular solutions : To find a simple particular solution, we may be able to guess and verify it.

36 3) Successive integration of two first-order equations

37 4) Exponential right-hand side First, suppose that c is not equal to either a or b. Solving the DE by the successive integration of two first-order equation gives the particular solution, e cx. ex) Backing to the previous DE,

38 5) Use of complex exponentials

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40 6) Method of undetermined coefficients The method of assuming an exponential solution and determining the constant factor C is an example of the method of undetermined coefficients. Example)

41 7) Several terms on the right-hand side; principle of superposition - Solve a separate equation and add the solutions.  principle of superposition (working only for linear equations)

42 8) Forced vibrations (steady state motion)

43 9) Resonance

44 10) Use of Fourier series in Finding particular solutions Example)

45

46 PROBLEM Solve the RLC circuit equation with V=0. Write the conditions and solutions for overdamped, critically damped, and underdamped electrical oscillations &

47 7. Other second-order equations

48 Example 1.(plus or minus sign much be chosen correctly at each stage of the motion so that the retarding force opposes the motion.) (describe the motion…)

49 Example)

50

51 Chapter 8 Ordinary differential equation Mathematical methods in the physical sciences 3rd edition Mary L. Boas Lecture 9 Laplace transform

52 8. The Laplace transform Example 1. f(t)=1 Example 2. f(t)=e^(-at) cf. Laplace transform are useful in solving differential equations. - Laplace transform

53 - Some properties of Laplace transform

54 Example 3. Let us verify L3. L(cos at)

55 Example 4. Let us verify L11. L(t sin at)

56 9. Solution of differential equations by Laplace transforms - Laplace transforms can reduce an linear DE to an algebraic equation and so simply solving it. Also since Laplace transforms automatically use given values of initial conditions, we find immediately a desired particular solution. Note) The relations already include the initial conditions.

57 Example 1.

58 Example 2.

59 Example 4.

Convolution - Method to write a formula for y Example 1. In this case, y is the inverse Fourier Transform of a product of two functions whose inverse transforms we know.

61 Example 2.

62 - Fourier Transform of a Convolution

63

The Dirac delta function - Impulse: impulsive force f(t), t=t_0 to t=t_1 - We are not interested in the shape of f(t). What we think important is the value of the integration of f(t) during t_1 – t_0 =  t.

65 - The above functions have the same integration value, 1.

66 - Laplace Transform of a  Function

67 - Example 2. - Example 3.

68 - Fourier Transform of a  Function

69 - Another physical application of  functions Example 4. 2 at x=3, -5 at x=7, and 3 at x=-4

70 - Derivative of the  function

71 - Some formulas involving  functions

72 -  functions in 2 or 3 dimensions Spherical coord. Cylindrical coord.

73 cf. divergence theorem

A brief introduction to Green Functions - Example 1