Module 13 - Differential Equations 3 (self Study)

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course

MODULE 13

DIFFERENTIAL EQUATIONS III

Module Topics 1. Linear operators 2. Linear inhomogeneous second order ordinary differential equations with constant coefficients 3. Free and forced oscillations

A:

Work Scheme based on JAMES (THIRD EDITION)

1. Linear and nonlinear differential equations were defined at the beginning of Module 6. A differential equation is linear when the dependent variable, or variables, and their derivatives do not occur as products, raised to powers or in nonlinear functions. There is a more formal way of defining linearity based on operators. Read the introduction to section 10.8 beginning on p.714 and then study section 10.8.1 including Examples 10.23-10.26. Differential operators provide a neat way of writing the general results on differential equations that are used in this Module. You should recall that it was pointed out in Module 6 that it is more usual to refer to equations of the type discussed near the bottom of p.716 as inhomogeneous (not nonhomogeneous). 2. Linear differential operators are used in section 10.8.2. Study the introductory paragraph and work through Example 10.27. Study the bottom half of p.717 and Examples 10.28 and 10.29. The stated linearity principle is extremely useful in determining general solutions to differential equations and was used in Module 6, for instance, in obtaining the general solutions to homogeneous second order linear ordinary differential equations with constant coefficients. Study the paragraph on p.718 above Example 10.30 and then Example 10.30 itself. The further idea needed is discussed in the two paragraphs below the horizontal line near the top of p.719. Linearly independent functions are ones that are essentially different. Linearly dependent functions are not completely distinct since one function can be obtained by taking a linear combination of some or all of the others. For example, sin t and cos t are linearly independent functions, but 1, t and 2 + 3t are dependent since the final function is a linear combination of the first two (2 + 3t = 2(1) + 3(t)). Continue studying pp.719 and 720 in J, and complete section 10.8.2 by working through Examples 10.31 and 10.32. The detailed results on pp.719 and 720 for the General solution of a linear homogeneous equation and the General solution of a linear nonhomogeneous (or inhomogeneous) equation, and the definitions of particular integral and complementary function in the middle of p.720, are all extremely important. ***Do Exercises 44(a), 47(c) on pp.721 and 722*** 3. The crucial part of this Module concerns linear inhomogeneous equations with constant coefficients. Turn to p.728, therefore, and study the introduction to section 10.9.3. The determination of particular integrals is not so difficult as suggested in J. and, as you will see below, there are straightforward methods available for the common forms of f (t). –1–

Study Examples 10.39-10.41. The crucial point to emphasise is that in solving inhomogeneous equations of the form L[x(t)] = f (t) the method is split into two parts - finding the complementary function, when the right-hand-side is zero (i.e. L[x(t)] = 0), and finding the particular integral, any solution however simple of the full equation (L[x(t)] = f (t)). Examples 10.39-10.41 contain the three most common types of f (t), i.e. polynomial, trigonometric and exponential. In each case it is not too difficult to spot likely trial functions for the particular integral, and the calculation of the particular integral is known as the method of undetermined coefficients, for obvious reasons. A summary of the commonly occurring trial functions is found in the table at the top of p.731. For linear ordinary differential equations with constant coefficients, solutions of the homogeneous equation L[x(t)] = 0 can be found using the method previously discussed in Module 6. Recall that the differential equation is replaced by an auxiliary equation in m, and from the solutions for m the general solution for x(t) can be written down. Examples 10.33-10.35 on p.725 illustrate the three different types of solution for m (2 distinct real roots, 2 equal roots and 2 complex roots), with the corresponding solutions of the ordinary differential equations, although the numerical answers for m1 and m2 in Example 10.33 are wrong. Before solving Exercises one further point needs clarification. In all of the Examples 10.39-10.41 J. calculates the particular integrals before finding the complementary functions. For reasons which will become apparent below, it is better to calculate the complementary function first and you are recommended to follow the latter procedure. ***Do Exercises 60(b), 61(a),(b) on p.734*** 4. If the function f (t) appearing in the equation L[x(t)] = f (t) is the sum of two terms of different type, e.g. t + sin t, then the trial particular integral is the sum of the trial functions corresponding to each term in the sum. Thus, when f (t) = t+ sin t, the trial function for the particular integral is At+ B + C sin t+ D cos t, since At + B and C sin t + D cos t are the trial functions for t and sin t respectively. Study the lower half of p.731, and then study Example 10.42. ***Do Exercise 61(h) on p.734*** 5. One further difficulty can arise when calculating particular integrals. Study the lower half of p.732 and then p.733, including Examples 10.43 and 10.44. As shown in these Examples the chosen trial function for the particular integral has had to be changed, because the expected form appears in the complementary function. With f (t) = et the usual trial function is x = Aet . However, this appears in the complementary function so the next try is x = Atet . Clearly this also occurs in the complementary function, so you must d2 x dx and and substituting into the original again multiply by t to get x = At2 et . After calculating dt dt2 ordinary differential equation, it is shown in J. that the constant A can be determined. ***Do Exercise 61(j) on p.734*** 6. The unknown constants which still appear in the general solution can be found from given boundary or initial conditions. These must be applied to the general solution, not to the complementary function only. ***Do Exercise 21(a) on p.762*** 7. There are a large number of engineering applications of linear second order ordinary differential equations with constant coefficients. To complete the module we will consider briefly some of these, but obviously most applications will arise in your engineering units. Turn to p.735 and read section 10.10.1. Then study section 10.10.2 on the free oscillations of damped elastic systems. Equations of the type (10.49) occur frequently in practice. For interpretative reasons and to make the algebra simpler it is helpful to write the basic equation as (10.50). You should remember how to calculate the parameters ω and ζ and how to establish whether a system is over-damped, under-damped or critically-damped. –2–

8. Read section 10.10.3 on the forced oscillations of elastic systems. The governing inhomogeneous equation can be written as (10.52), or more simply as (10.53). The trial particular integral for (10.53) is d2 x dx and , substituting into 10.53 and equating the coefficients x = A cos Ωt+B sin Ωt. After calculating dt dt2 of cos Ωt and sin Ωt, the coefficients A and B can be determined. The particular integral then has the form (10.54a). Since cos(Ωt − δ) = cos Ωt cos δ + sin Ωt sin δ, it is possible to compare again the coefficients of cos Ωt and sin Ωt and deduce that the particular integral has the alternative expression (10.54b). It is important to compare and contrast the original forcing term on the right-hand-side of Equation (10.53) with the steady-state solution (10.54b). The solution has the same frequency as the forcing term, its amplitude is multiplied by the factor A(Ω) and it is out-of-phase with the forcing term because δ in (10.54b) is usually non-zero. The phenomenon of resonance obviously occurs if A takes a large value. The mathematics underlying section 10.10.3 is exactly the same as that in section 10.10.4 on oscillations in electrical circuits. ***Do Exercises 64(b), 66(d) on p.747***

B:

Work Scheme based on STROUD (FIFTH EDITION)

Linear operators are not mentioned in S. For this small, and less important, topic you should work through sections 1 and 2 of the above work scheme based on JAMES (THIRD EDITION). The main part of the Module is covered in Programme 25 of S, starting on p.1081. Work through frames 23-49. For the final small topic in the Module you should return to the work scheme based on J, and go through sections 7 and 8.

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Specimen Test 13 1.

Write down the differential operator L that allows the following differential equation to be expressed as L[x(t)] = 0: d2 x dx + 4x = 0 . +3 2 dt dt

2.

Consider the differential equation

d2 x dx + 6x = f (t) . +5 dt2 dt

(i) Show that the complementary function is x = Ae−3t + Be−2t . (ii) Find a particular integral of the differential equation when (a)

f (t) = cos t,

(b)

f (t) = e−2t ,

(c)

f (t) = 1 − 6t.

(iii) Using the above results, where appropriate, write down the general solution of the equation dx d2 x + 6x = e−2t , +5 dt2 dt and hence derive the solution which satisfies the conditions x =

dx 1 and = 0 when t = 0. 4 dt

3. (i) Find the damping parameter and natural frequency of the system governed by the equation dx d2 x + 16x = 0 , + 6α dt2 dt given that α is a positive constant. (ii) For what values of α is the above equation (a) underdamped,

(b) critically damped?

(iii) State (without solving the differential equation) which parts of the general solution describe the transient motion and steady state solution respectively.

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