2. Applications of Second Order Differential Equations : Vibration of Spring © Dr. Elmer P. Dadios - DLSU Fellow & Professor Thus the general solution is: Equation 1 is a second order linear differential equation. Its Auxiliary equation is: Which can also be written as: Where:

3. Applications of Second Order Differential Equations : Vibration of Spring © Dr. Elmer P. Dadios - DLSU Fellow & Professor Solution: from Hookes’ law the force required to lift the spring: Using this value of k and m = 2 in equation 1, we have: The solution is: 2

4. Applications of Second Order Differential Equations : Vibration of Spring © Dr. Elmer P. Dadios - DLSU Fellow & Professor The solution is:

5. Applications of Second Order Differential Equations : Vibration of Spring © Dr. Elmer P. Dadios - DLSU Fellow & Professor or: Auxiliary equation is: Roots are : r 1 = -4 and r 2 = -16 Solution is:

6. Applications of Second Order Differential Equations : Vibration of Spring © Dr. Elmer P. Dadios - DLSU Fellow & Professor so: Therefore:

7. Forced Vibrations © Dr. Elmer P. Dadios - DLSU Fellow & Professor Thus: 5

8. Forced Vibrations © Dr. Elmer P. Dadios - DLSU Fellow & Professor A commonly occurring type of external force is a periodic force function. 6 In the absence of a damping force (c=0) we can use the method of undetermined coefficients to show that.

9. Electrical Circuits © Dr. Elmer P. Dadios - DLSU Fellow & Professor Fig 7 contains an electromotive force E (supplied by a battery or generator), a resistance R, an inductor L, and a capacitor C, in series.

10. Electrical Circuits © Dr. Elmer P. Dadios - DLSU Fellow & Professor If the charge on the capacitor at time t is Q = Q(t), then the current is the rate of change of Q with respect to t: I = dQ/dt. The voltage drops across the resistor, inductor, and capacitor is: 7 KVL says that the sum of these voltage drops is equal to the supplied voltage. Since

11. Electrical Circuits © Dr. Elmer P. Dadios - DLSU Fellow & Professor Equation 7 is a second order linear differential equation with constant coefficients. If the charge Q(0) and current I(0) is known at time 0 then we have the initial condition: Differentiating equation 7 noting that:

12. Electrical Circuits © Dr. Elmer P. Dadios - DLSU Fellow & Professor 8 With the given values of R, L, C, and E(t) from equation 7 Auxiliary equation is: Root is:

13. Electrical Circuits © Dr. Elmer P. Dadios - DLSU Fellow & Professor Then Substituting into equation 8 Hence the solution of complementary equation is For the method of undetermined coefficients the particular solution is:

14. Electrical Circuits © Dr. Elmer P. Dadios - DLSU Fellow & Professor

15. Electrical Circuits © Dr. Elmer P. Dadios - DLSU Fellow & Professor

16. Electrical Circuits © Dr. Elmer P. Dadios - DLSU Fellow & Professor

17. Electrical Circuits © Dr. Elmer P. Dadios - DLSU Fellow & Professor So for large value of t For this reason Q p (t) is called steady state solution Below shows how the graph of steady state solution compares with the graph of Q

18. Comparison © Dr. Elmer P. Dadios - DLSU Fellow & Professor Mathematically 5 and 7 are identical

19. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor Many differential equations can’t be solved explicitly in terms of finite combinations of simple familiar functions. This is true even for a simple looking equation like: 1 In this case, we use the method of power series: we look for a solution of the form The method is to substitute this expression into the differential equation and determine the values of the coefficients c 0, c 1, c 2, etc.

20. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor Example 1: Use power series to solve 2 We can differentiate the power series term by term Solution: We assume there is a solution of the form 3

21. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor 4 Substituting the expression in 2 and 4 into the differential equation. 5 or

22. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor 6

23. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor By now we see the pattern ? ? ?

24. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor Pattern is:

25. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor

26. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor We recognize the series obtained as being the Maclaurin series for cosx and sinx. Therefore we could write the solution as:

27. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor Example 2: Solve the equation: Solution : We assume that there is a solution of the form Then:

28. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor Substitute to our equation

29. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor Thus: 7 We solve this recursion by putting n = 0, 1, 2, 3,... successively

30. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor Thus:

31. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor Thus:

32. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor In General: the even coefficient is: the odd coefficient is:

33. Therefore the Solution is: © Dr. Elmer P. Dadios - DLSU Fellow & Professor Or:

34. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor Note: if we are asked to solve the initial value problem We would observe that: This would simplify the calculation in this example, since all of the even coefficients would be 0. The solution to the initial value problem is:

35. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor Unlike the situation of example 1, the power series that arise in the solution do not define elementary functions. The functions above are perfectly good functions but they can’t be expressed in terms of familiar functions. We can use these power series expressions for y 1 and y 2 to compute approximate values of the functions and even to graph them.

36. Series Solution © Dr. Elmer P. Dadios - DLSU Fellow & Professor Figure below shows the first few partial sums T 0, T 2, T 4, …. Taylor polynomials for y 1 (x), and see how they converge to y 1. See the second graph.

37. Complex variable © Dr. Elmer P. Dadios - DLSU Fellow & Professor A complex number has a real part and an imaginary part, both of which are constant. If the real part or the imaginary part (or both) are variables, then complex number is called a complex variable. In the Laplace transformation, we use the notation s to denote a complex variable; that is, s = σ + jω where σ is the real part and jω is the imaginary part Note that both σ and ω are real.

38. Complex variable © Dr. Elmer P. Dadios - DLSU Fellow & Professor Complex function. A complex function F(s), has a real part and an imaginary part Fs = F x + jF y Where F x and F y are real quantities. The Magnitude of F(s) is: The angle θ of Fs is: measured counterclockwise from the positive real axis The complex conjugate of F(s) is:= F x - jF y

39. Complex Functions © Dr. Elmer P. Dadios - DLSU Fellow & Professor Complex functions commonly encountered in linear systems analysis are single valued functions of s and are uniquely determined for a given value of s. Typically: Points at which F(s) equals zero are called zeros. That is s = -z1; s = -z2 …. s = -zm Points at which F(s) equals infinity are called poles That is s = -p1, s = -p2 …s = - pm Note that F(s) may have additional zeros at infinity

40. Complex Functions © Dr. Elmer P. Dadios - DLSU Fellow & Professor If the denominator of F(s) involves k-multiple factors Then s = -p is called a multiple pole of order k or repeated pole of order k. If k = 1 then the pole is called a simple pole.

41. Complex Functions © Dr. Elmer P. Dadios - DLSU Fellow & Professor Example: consider the complex function G(s) has zeros at s = -2 and s = -10 Simple poles at s = 0, s = -1, s = -5, Double pole at s = -15 Note that G(s) becomes zero at s = ∞ Since, for large values of s, It follows that G(s) possesses a triple zero (multiple zero of order 3) at s = ∞

42. Complex Functions © Dr. Elmer P. Dadios - DLSU Fellow & Professor s = -2, s = -10, s = ∞, s = ∞, s = ∞ If points at infinity are included, G(s) has the same number of poles as zeros. To summarize, G(s) has five zeros at And five poles at s = 0, s = -1, s = -5, s = -15, s = -15

43. Complex Functions © Dr. Elmer P. Dadios - DLSU Fellow & Professor s-Domain equivalent circuits and impedances

44. Laplace Transform © Dr. Elmer P. Dadios - DLSU Fellow & Professor The Laplace transform is one of the most important mathematical tools available for modeling and analyzing linear systems. The Laplace transform method is an operational method that can be used advantageously in solving linear, time-invariant differential equations. Its main advantage is that differentiation of the time function corresponds to multiplication of the transform by a complex variable s, and thus the differential equations in time become algebraic equations in s. The solution of the differential equation can then be found by using a Laplace transform table or the partial-fraction expansion technique. Another advantage of the Laplace transform method is that, in solving the differential equation, the initial conditions are automatically taken care of, and both the particular solution and the complementary solution can be obtained simultaneously.

45. Laplace Transform © Dr. Elmer P. Dadios - DLSU Fellow & Professor In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems.mathematicslinear time-invariantsystemselectrical circuitsharmonic oscillatorsoptical devices Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications. The Laplace transform is an important concept from the branch of mathematics called functional analysis.functional analysis

46. Laplace Transform © Dr. Elmer P. Dadios - DLSU Fellow & Professor In actual physical systems the Laplace transform is often interpreted as a transformation from the time-domain point of view, in which inputs and outputs are understood as functions of time, to the frequency-domain point of view, where the same inputs and outputs are seen as functions of complex angular frequency, or radians per unit time.time-domainfrequency-domaincomplexangular frequencyradians This transformation not only provides a fundamentally different way to understand the behavior of the system, but it also drastically reduces the complexity of the mathematical calculations required to analyze the system. The Laplace transform has many important applications in physics, optics, electrical engineering, control engineering, signal processing, and probability theory.physicsoptics electrical engineeringcontrol engineeringsignal processingprobability theory The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace, who used the transform in his work on probability theory.mathematicianastronomer Pierre-Simon Laplaceprobability theory

47. Laplace Transform © Dr. Elmer P. Dadios - DLSU Fellow & Professor The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by:functionreal numbers The parameter s is in general is complex:complex The lower limit of 0 − is short notation to mean This integral transform has a number of properties that make it useful for analyzing linear dynamical systems.integral transformdynamical systems The most significant advantage is that differentiation and integration become multiplication and division, respectively, with s.differentiationintegration (This is similar to the way that logarithms change an operation of multiplication of numbers to addition of their logarithms.) This changes integral equations and differential equations to polynomial equations, which are much easier to solve.logarithmsintegral equations differential equationspolynomial equations

48. Laplace Transform © Dr. Elmer P. Dadios - DLSU Fellow & Professor The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by:functionreal numbers The parameter s in general is complex:complex The lower limit of 0 − is short notation to mean This integral transform has a number of properties that make it useful for analyzing linear dynamical systems.integral transformdynamical systems The most significant advantage is that differentiation and integration become multiplication and division, respectively, with s.differentiationintegration (This is similar to the way that logarithms change an operation of multiplication of numbers to addition of their logarithms.) This changes integral equations and differential equations to polynomial equations, which are much easier to solve.logarithmsintegral equations differential equationspolynomial equations

49. Bilateral Laplace Transform © Dr. Elmer P. Dadios - DLSU Fellow & Professor When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is normally intended. The Laplace transform can be alternatively defined as the bilateral Laplace transform or two-sided Laplace transform by extending the limits of integration to be the entire real axis.two-sided Laplace transform If that is done the common unilateral transform simply becomes a special case of the bilateral transform where the definition of the function being transformed is multiplied by the Heaviside step function (also known at the unit step function).Heaviside step function The bilateral Laplace transform is defined as follows:

50. Inverse Laplace Transform © Dr. Elmer P. Dadios - DLSU Fellow & Professor The inverse Laplace transform is the Bromwich integral, which is a complex integral given by:Bromwich integralcomplex where γ is a real number so that the contour path of integration is in the region of convergence of F(s) normally requiring γ > Re(s p ) for every singularity s p of F(s) and i 2 =-1. region of convergence singularity If all singularities are in the left half-plane, that is Re(s p ) < 0 for every s p, then γ can be set to zero and the above inverse integral formula above becomes identical to the inverse Fourier transform.Fourier transform

51. Region of convergence © Dr. Elmer P. Dadios - DLSU Fellow & Professor The Laplace transform F(s) typically exists for all complex numbers such that Re{s} > a, where a is a real constant which depends on the growth behavior of f(t), whereas the two-sided transform is defined in a range a < Re{s} < b. The subset of values of s for which the Laplace transform exists is called the region of convergence (ROC) or the domain of convergence. In the two-sided case, it is sometimes called the strip of convergence. There are no specific conditions that one can check a function against to know in all cases if its Laplace transform can be taken, other than to say the defining integral converges.

52. Properties and theorems © Dr. Elmer P. Dadios - DLSU Fellow & Professor Given the functions f(t) and g(t), and their respective Laplace transforms F(s) and G(s): See tables.

53. Laplace Transform: Examples © Dr. Elmer P. Dadios - DLSU Fellow & Professor Example 1. Deriving the complex impedance for a capacitor This example is based on the principles of electrical circuit theory.electrical circuit The constitutive relation governing the dynamic behavior of a capacitor is the following differential equation:capacitor Where: C is the capacitance (in farads) of the capacitor, I = i(t) is the electrical current (in amperes) flowing through the capacitor as a function of time, v = v(t) is the voltage (in volts) across the terminals of the capacitor, also as a function of time.farads electrical currentamperesvoltagevolts

54. Laplace Transform: Examples © Dr. Elmer P. Dadios - DLSU Fellow & Professor Taking the Laplace transform of this equation, we obtain where, Solving for V(s) we have

55. Laplace Transform: Examples © Dr. Elmer P. Dadios - DLSU Fellow & Professor, The definition of the complex impedance Z (in ohms) is the ratio of the complex voltage V divided by the complex current I while holding the initial state V o at zero:compleximpedanceohms Using this definition and the previous equation, we find: which is the correct expression for the complex impedance of a capacitor.

56. Laplace Transform: Examples © Dr. Elmer P. Dadios - DLSU Fellow & Professor, Example 2: Method of partial fraction expansion Consider a linear time-invariant system with transfer functiontransfer function The impulse response is simply the inverse Laplace transform of this transfer function:impulse response To evaluate this inverse transform, we begin by expanding H(s) using the method of partial fraction expansionpartial fraction expansion

57. Laplace Transform: Examples © Dr. Elmer P. Dadios - DLSU Fellow & Professor, Example 2: Method of partial fraction expansion for unknown constants P and R. To find these constants, we evaluate and

58. Laplace Transform: Examples © Dr. Elmer P. Dadios - DLSU Fellow & Professor, Example 2: Method of partial fraction expansion Substituting these values into the expression for H(s), we find Finally, using the linearity property and the known transform for exponential decay (see Item #3 in the Table of Laplace Transforms, above), we can take the inverse Laplace transform of H(s) to obtain: which is the impulse response of the system.