Chapter 7, Sections 7.1, 7.2, 7.3, 7.4, 7.5 Topics, Definitions, Theorems Section 7.1 Laplace Transforms and Inverse Transforms =========== The Laplace Transform solves differential equations. It should be advertised as the METHOD of QUADRATURE for Nth ORDER ODE and SYSTEMS of DIFFERENTIAL EQUATIONS. DEF. Laplace transform. Direct Laplace transform. Laplace integral. DEF. Improper integral on (0,infinity). Converge. Diverge. Assume always in Laplace theory that an input satisfies f(t)=0 for t<0, and f(t) is defined by a t-expression for t>=0. Assume always in Laplace theory that a Laplace integral is an s-expression defined for all large s. EXAMPLE 1. Laplace integral of f(t)=1. ANSWER: L(1)=1/s EXAMPLE 2. Laplace integral of f(t)=exp(at). ANSWER: L(exp(at))=1/(s-a) DEF. Generalized factorial function. Gamma function Gamma(x)=int(G(t,x),t=0..infinity) where G(t,x)=t^(x-1) exp(-t), for x>0 and 00, Gamma(1)=1, Gamma(n+1)=n! for positive integers n. Gamma(1/2)=sqrt(Pi) [page 579] EXAMPLE 3. Laplace integral of f(t)=t^a, a > -1 real. ANSWER: L(t^a)=Gamma(a+1)/s^(a+1) LEMMA. L(t)=1/s^2, L(t^2)=2/s^3, L(t^3)=6/s^4 PROOF. See the lecture slides. THEOREM 1. Linearity of the Laplace transform. L(af+bg)=aL(f)+bL(g) EXAMPLE 4. Laplace integral of f(t)=t^(n/2), L(3t^2+4t^(3/2)) ANSWER: L(3t^2+4t^(3/2))=6/s^3+3 sqrt(Pi)/s^(5/2) The basic important results are for exp(at), cos(kt), sin(kt). EXAMPLE 5a. Compute L(cosh(kt)) and L(sinh(kt)) ANSWER: s/(s^2-k^2) and k/(s^2-k^2), resp. EXAMPLE 5b. Compute L(cos(kt)) and L(sin(kt)) ANSWER: s/(s^2+k^2) and k/(s^2+k^2), resp. EXAMPLE 6. Compute L(3e^(2t) + 2 sin^2(3t)) ANSWER: Use identity cos(2 theta)=1 - 2 sin^2(theta) L(3e^(2t) + 2 sin^2(3t))=3/(s-2) + 1/s - s/(s^2+36) INVERSE TRANSFORM DEF. L^(-1)(F(s))=f(t) means L(f(t))=F(s). Makes sense because of Lerch's theorem, Theorem 3 below. CANCELLATION LAW. If L(f(t))=L(g(t)), then f(t)=g(t). The hypothesis requires "for large s" and the conclusion "at points of continuity". EXAMPLE 7. Find the inverse transform of 7a. 1/s^3 7b. 1/(s+2) 7c. 2/(s^2+9) ANSWERS: (a) t^2/2, (b) exp(-2t), (c) (2/3) sin(3t) WORDS "Find the function f(t) whose transform is 1/s^3" "Find the inverse transform of 1/s^3" "Read the Laplace table backwards" FIGURE 7.1.2 Short table of transforms Transform table too long. Too many extras. It is enough to have entries 1,2,3,5,6,7. Then a second table, of lessor-used transforms. The words "forward table" and "backward table" summarize what readers have to learn, in order to use the Laplace table. Page 581. DEF. Unit step u(t) The table entry for u(t-a) should say a>=0. MISSING: the unit pulse definition u(t-a)-u(t-b). Applications in 7.1 exercises 7, 8, 9, 37, 38, 40, 41, 42. Page 581 PIECEWISE CONTINUOUS FUNCTIONS The precise mathematical definition with limits is needed. Engineering use is primarily in terms of superposition of pulses multiplied by shape functions. f(x) = sum of terms v[n] for n=1 to N v[n]= continuous function times a pulse DEF. unit pulse = (t,a,b) --> u(t-a)-u(t-b), for a0, a>0 Should be a>=0. See also Table 7.1.2, where a>=0 is missing. DEF. Function of exponential order. Page 583 THEOREM 2. Existence of L(f(t)). COROLLARY. Limit of F(s) as s --> infinity. Page 584 THEOREM 3. Lerch's theorem, L(f)=L(g) implies f=g. ==================================================================== Page 587 7.1 APPLICATION: CAS ===== ==================================================================== 7.2 Transformation of Initial Value Problems ===== Page 588 THEOREM 1. If f is continuous and piecewise smooth and of exponential order, then L(f')=sL(f)-f(0). The alternate definition of piecewise smooth is that f(x) is a continuous function expressed as the sum of terms g(x) times a unit pulse, where g(x) is smooth on -infinity to infinity. Could the alternate wording appear, along with a displayed equation? FIGURE 7.2.1. Discontinuities of f' Page 589 COROLLARY. Transforms of Higher Derivatives EXAMPLE 1. Solve x''-x'-6x=0, x(0)=2, x'(0)=-1 ANSWER: x(t)=(1/5)(3 exp(3t) + 7 exp(-2t)) DETAILS. Transform by (4) and (5) rules, isolate X(s) left. Solve the partial fraction problem. Use the Laplace table backwards. Apply Lerch's theorem or take the inverse transform, to find x(t). Page 590 REMARK. Partial fraction methods. Missing is Heaviside's cover-up method. EXAMPLE 2. Solve x'' + 4x = sin(3t), x(0)=0, x'(0)=0 ANSWER: x(t)=(1/10)(3 sin(2t) - 2 sin(3t)) DETAILS. Uses equation (5) and the shortcut for zero initial conditions. Then X(s) is isolated left to give a partial fraction problem with quadratic factors. Shortcut: Using (4) and (5) in the case of zero conditions is much easier. Answer checks along the route to isolating X(s) left include looking for the characteristic equation: a preliminary sanity check. FIGURE 7.2.4. Flow chart for solving a DE by Laplace. Page 592 LINEAR SYSTEMS EXAMPLE 3. Solve the system 2x''=-6x + 2y, y''=2x - 2y + 40 sin(3t), x(0)=x'(0)=y(0)=y'(0)=0. ANSWER: x(t)=5 sin(t) - 4 sin(2t) + sin(3t), y(t)=10 sin(t) + 4 sin(2t) - 6 sin(3t) DETAILS. Transform the system, get a 2x2 system for X(s), Y(s). Solve by Cramer's rule and expand in partial fractions. Find the inverse transforms of X(s), Y(s) from the Laplace table. Page 593 TRANSFORM PERSPECTIVE. ==> The background is sections 5.1 to 5.4. Please alter the first sentence of the subsection to insert the back-reference to section 5.4. EXAMPLE 3.5. Transform of a forced damped oscillator. X(s) = F(s)/Z(s) + I(s)/Z(s), 1/Z(s)=transfer function EXAMPLE 4. Compute L(t exp(at)) and verify it equals 1/(s-a)^2. EXAMPLE 5. Compute L(t sin(kt)) DETAILS. Use (5), f(t)=t sin(kt) and the facts f(0)=f'(0)=0. ANSWER: F(s)=2ks/(s^2+k^2)^2 THEOREM 2. Transforms of Integrals Let f be piecewise continuous of exponential order. Then L(integral of f on [0,t]) = F(s)/s EXAMPLE 6. Find the inverse transform of G(s)=(1/s^2)/(s-a) ANSWER: (1/a^2)(exp(at)-at-1) DETAILS: Use the integral theorem twice. Page 596 Proof of theorem 1. Page 597 Extension of theorem 1. Page 598 EXAMPLE 7. Let f(t)=1+floor(t), the unit staircase. Apply the extension to theorem 1, to obtain the identity L(f)=(1/s)/(1-exp(-s)). Page 598 EXERCISES 7.2 Exercises 11 to 16 require background reading from section 7.1, especially Picard's theorem on page 403. Exercise 16 should have a footnote to 4.1-26 (brine tanks) and compartment analysis section 5.3. ==================================================================== 7.3 Translation and Partial Fractions ===== DEF. A partial fraction is a constant divided by a polynomial with exactly one root (real or complex). DEF. A quadratic partial fraction is a linear polynomial divided by a polynomial with exactly one pair of complex conjugate roots. Page 600 The total number of terms in the complete partial fraction expansion produced by (2) and (3) can be predicted from the degree of Q(x). "The degree of Q counts the number of terms in the partial fraction expansion of P(x)/Q(x)." Page 601 THEOREM 1. Translation on the s-axis [First shifting theorem] If L(f) exists then so does L(exp(at)f(t)) and L(exp(at)f(t))=F(s-a). TABLE page 601 Equations (6), (7), (8) are important. EXAMPLE 1. Spring-mass system with damping, no forcing. Solve (1/2)x'' + 3x' + 17x=0, x(0)=3, x'(0)=-1. ANSWER: x(t)=exp(3t)(3 cos(5t) + 2 sin(5t)) DETAILS. Transform using (4) and (5) from 7.2. Isolate X(s) left, expand right side s-fraction into partial fractions with quadratic factors. Find inverse transforms, report the answer x(t). Page 602 EXAMPLE 2. Find the inverse transform of R=(s^2+1)/(s^3-2s^2-8s) ANSWER: -1/8 + (5/12)exp(-2t) + (17/24)exp(4t) DETAILS. Partial fractions with simple roots. EXAMPLE 3. Solve y''+4y'+4y=t^2, y(0)=y'(0)=0 ANSWER: y(t)=t^2/4 - t/2 + 3/8 - t exp(-2t)/4 - 3 exp(-2t)/8 DETAILS. Transform using (4), (5) in 7.2, isolate Y(s) left, with right side fraction R = (2/s^3)/(s+2)^2. Degree of the denominator is 5, need 5 partial fractions in the expansion, which involves repeated roots. The technique is longer than sampling the fraction-free equation with 5 samples (s=0,s=2, and 3 invented samples) to obtain a 5x5 linear algebraic system of equations for A to E. Page 604 EXAMPLE 4. Spring-mass system with forcing term. Solve x''+6x'+34x = 30 sin(2t), x(0)=x'(0)=0. ANSWER: x(t) = (5/29)(-2 cos(2t) + 5 sin(2t)) [periodic steady-state] +(2/29)exp(-3t)(5 cos(5t) - 2 sin(5t)) [transient] DETAILS. Transform using (4), (5) from 7.2, isolate X(s) left and on the right side the fraction R = (60/(s^2+4)) / ((s+3)^2+25). Expand R into 2 quadratic partial fractions, with constants A, B, C, D. Clear fractions. Substitute s^2+1=0 to get a 2x2 system for A.B. Solve A=-10/29, B=50/29. Substitute (s+3)^2+25=0 to get a 2x2 system for C,D. Solve for C=D=10/29. Take inverse transforms. Missing detail in the example. Does not show how to arrange the fractions in order to use the Laplace table. Most would use the shifting theorem first on the fraction with (s+3)^2+25. Page 606 RESONANCE and REPEATED QUADRATIC FACTORS Formulas (16), (17) for fractions with the square of a quadratic in the denominator. EXAMPLE 5. Undamped forced oscillations, spring-mass system. Solve x'' + omega0^2 x = F0 sin(omega0 t), x(0)=x'(0)=0 ANSWER: x(t) = (1/2)(F0/omega0^2)(sin(omega0 t) - omega0 t cos(omega0 t)) DETAILS. Transform using (5) in 7.2, isolate X(s) left, and on the right side is fraction R=F0 omega0 /(s^2+omega0^2)^2. Apply (17). FIGURE 7.3.4. Graphic of x(t) and envelope curves. EXAMPLE 6. Solve (D^4+2D^2+1)y=4 t exp(t), y(0)=y'(0)=y''(0)=y'''(0)=0 ANSWER: y(t) = (t-2) exp(t) + (t+1) sin(t) + 2 cos(t) DETAILS. Transform by (5), (7) in 7.2, use zero initial data shortcut. Isolate Y(s) left, with right side fraction R=(4/(s-1)^2)/(s^2+1)2 The denominator of R has degree 6, the partial fraction expansion has 6 constants A to F, with 2 partial fractions and two quadratic partial fractions. Clear fractions. Determine A=1 using s=1. Insert 5 samples s=0, -1, 2, -2, 3 to get a 5x5 system for B to F. Solve it for B=-2, C=2, D=0, E=2, F=1. Use Laplace tables to finish. Page 608 EXERCISES 7.3 ==================================================================== 7.3 APPLICATION: Damping and Resonance ===== A good reference for maple commands. CAS with maple is used in a long maple lab involving a variety of engineering functions, periodic extensions and solutions of initial value problems. ==================================================================== 7.4 Derivatives, Integrals and Products of Transforms ===== Page 610 EXAMPLE 1a. Show that L(cos(t) sin(t) does not equal L(cos(t)) L(sin(t)). DEF. Convolution f*g For completeness, an exercise can be added which relates the convolution of two functions f, g on -infinity to infinity to f*g. This uses the assumption that f, g are zero for t<0, in Laplace theory. EXERCISE. Define =integral of f(x-t)g(t) over t=-inf to inf. Assume f and g are zero for t<0. Show that =f*g. This verifies that the convolution defined in Fourier theory agrees with f*g, provided f=g=0 for t<0. Page 611 EXAMPLE 1. Verify cos(t)*sin(t) = (1/2) t sin(t) THEOREM 1. CONVOLUTION PROPERTY Let f, g be piecewise continuous and of exponential order. Then f*g is defined and L(f*g) =L(f)L(g) Page 612 EXAMPLE 2. Find sin(2t) * exp(t) and use the answer to compute the inverse transform of the fraction R=(2/(s^2+4))/(s-1). ANSWER: L(sin(2t)=2/(s^2+4) L(exp(t))=1/(s-1) L(sin(2t)*exp(t))=(2/5) exp(t) - (1/5) sin(2t) - (2/5) cos(2t) inverse laplace of R = t-expression on previous line Missing in the example are the two equations L(sin(2t))=2/(s^2+4) and L(exp(t))=1/(s-1), essential to understanding the details. The purpose of the example is to compute the Laplace inverse of the fraction R. THEOREM 2. Differentiation of Transforms [s-diff theorem] If f is piecewise continuous and of exponential order, then L(-t f(t)) = F'(s) L((-t)^n f(t)) = (d/ds)^n F(s) EXAMPLE 3. Find L(t^2 sin(kt)) ANSWER: (6ks^2-2k^3)/(s^2+k^2)^2 DETAILS. Apply Theorem 2 to get F(s)=(-1)^2 (d/ds)^2 (k/(s2+k^2). Rest is calculus. EXAMPLE 4. Find the inverse transform of F(s)=arctan(1/s) ANSWER: f(t)=sin(t)/t DETAILS. Apply Theorem 2, L((-t)f(t)) = (d/ds) F(s) to get L(-t f(t)) = -1/(s^2+1), which equals L(-sin(t)). Then the cancellation law implies -t f(t) = -sin(t). EXAMPLE 5. Find X(s)=L(x(t)) by Laplace's method for Bessel's equation t x'' + x' + t x = 0 with x(0)=1 and x'(0)=0. ANSWER: X=1/sqrt(s^2+1). DETAILS. Laplace's method gives (s^2+1)X'(s)+X(s)=0, which is a separable differential equation with solution X=C/sqrt(s^2+1). C=1 obtained in an exercise. Page 614. INTEGRATION OF TRANSFORMS THEOREM 3. Integration of transforms Assume f piecewise continuous of exponential order and f(t)/t has a finite right limit at t=0. Then L(f(t)/t) = integral of F(s) over [s,infinity) EXAMPLE 6. Let f(t)=sinh(t). Find L(f(t)/t). ANSWER: L(f(t)/t)=(1/2)ln|s+1|-(1/2)ln|s-1| DETAILS. Check the right limit at t=0 for f(t)/t using L'Hopital's rule. Apply Theorem 3. Use L(sinh(t)) = (1/2)/(s-1)-(1/2)/(s+1). Then integrate to get logs. We have to write the difference of logs as the log of a quotient in order to evaluate at s=infinity (ln(1)=0). EXAMPLE 7. Find the inverse Laplace transform of F(s)=2s/(s^2-1)^2. ANSWER: f(t)= t sinh(t). DETAILS. Apply Theorem 3, L(f(t)/t)==integral of F(s) over [s,infinity). The integration of F(s) gives 1/(s^2-1)=L(sinh(t)). Then L(f(t)/t)=L(sinh(t)), and cancellation implies f(t)/t = sinh(t). Page 615. PROOFS OF THE THEOREMS Try to read pages 615-616. The proofs of theorems 2, 3 are usually done in class. ==================================================================== Page 617 EXERCISES Problems 1-34 are recommended. The remaining problems are quite challenging. ==================================================================== 7.5 Periodic and Piecewise Continuous Input Functions ===== Page 618 DEF. unit step. Repeated from 7.1. THEOREM. L(u(t-a)) = exp(-as)/s for a>=0, s>0. THEOREM 1. Translation on the t-axis [second shift theorem] (a) If L(f) exists for s>c and a>=0, then for s>c+a L(u(t-a)f(t-a)) = exp(-as) F(s) (b) If a>=0 and L(g(t+a)) exists for large s, then L(u(t-a)g(t)) = exp(-as) L(g(t+a)) Error: Page 618, statement of theorem 1, missing a>=0. Missing: Page 618, statement of theorem 1, missing part (b) of the second shifting theorem. Part (b) is suited for finding the transform of terms like sin(t) u(t-2) whereas part (a) is suited for inverse transforms of terms like exp(-2s)/(s^2+4). Parts (a) and (b) have an alternative notation, which suggests the translations explicitly, but also they are easier to use on a given problem, due to looking like the calculus u-substitution. (a) If L(f) exists for s>c and a>=0, then for s>c+a L(u(x)f(x)|x=t-a) = exp(-as) F(s) (b) If a>=0 and L(g(x)|x=t+a) exists for large s, then L(u(t-a)g(t)) = exp(-as) L(g(x)|x=t+a) Example: L(sin(t)u(t-Pi)) = exp(-Pi s) L(sin(x)|x=t+Pi) Page 619 Error: Proof of theorem 1. Missing a>=0, used when splitting the integral over [0,a] and [a,infinity). EXAMPLE 1. Find the inverse transform of exp(-as)/s^3. ANSWER: (1/2)(t-a)^2u(t-a) for a>=0 DETAILS. Using Theorem 1 part (a), exp(-as)L(f(t))= L(u(t-a)f(t-a)) [backwards on purpose] and comparing to exp(-as)/s^3, choose L(f(t))=1/s^3 or f(t)=t^2/2. Then exp(-as)/s^3 = exp(-as)L(t^2/2)= L(x^2/2 u(x)|x=t-a), giving the answer x^2/2 u(x)|x=t-a = (1/2)(t-a)^2 u(t-a). EXAMPLE 2. Find L(g(t)) if g(t)=t^2u(t-3). ANSWER: exp(-3s)(2/s^3 + 6/s^2 + 9/s) DETAILS. Using theorem 1, part (b), L(t^2 u(t-3))=exp(-3s)L(x^2|x=t+3)= exp(-3s)L((t+3)^3). Expanding the square and using the Laplace table gives the result. Page 620 EXAMPLE 3. Pulse problem. Find L(f(t)), given f(t) = cos(2t) pulse(t,0,Pi) ANSWER: s(1-exp(-2 PI s))/(s^2+4) DETAILS. Write the pulse as 1-u(t - 2 Pi). Then f(t)=cos(2t) - cos(2t)u(t - 2 Pi) Replace cos(2t) by the equivalent expression cos(2t - 2 Pi) in the second term. Then use the second shifting theorem as follows L(f(t))=L(cos(2t)) - L(cos(x)u(x)|x=2t-2Pi) =L(cos(2t)) - exp(-2Pi s)L(cos(2t)) = (1-exp(-2Pi s))(2/(s^2+4)) EXAMPLE 4. Forced harmonic oscillator with pulse force. Solve x'' + 4x = f(t), x(0)=x'(0)=0, where f(t)=cos(2t) pulse(t,0,2Pi) ANSWER: x(t) = (1/4) t sin(2t) for t < 2Pi, f(t) = (Pi/2)sin(2t) for t>= 2Pi. DETAILS. Transform the differential equation by (5) in section 7.2. Isolate X(s) left and on the right expand L(f(t)) using the answer from Example 3. Then X(s)=s/(s^2+4)^2 - exp(-2Pi s)(s/(s^2+4)^2). Use the Laplace table on the inside cover of the book to obtain the formula L((t/4)sin(2t))=s/(s^2+4)^2. Use the second shifting theorem on the second term in X(s), as follows: exp(-2Pi s)(s/(s^2+4)^2) = L((x/4)sin(2x)u(x)|x=t-2Pi) = L(((t-2Pi)/4)sin(2(t-2Pi))u(t-2Pi)) The last challenge is to write this piecewise defined function as a normal display. Page 621 TRANSFORMS of PERIODIC FUNCTIONS THEOREM 2. Periodic Function Theorem Let f(t) satisfy f(t+p)=f(t), where p>0 is the period of f, and assume f is piecewise continuous for t>=0. Then L(f(t)) exists and is given by the equation L(f(t)) = P(t)/(1-exp(-ps)), P(t) = integral of f(t)exp(-st) over t=0 to t=p. Page 622 Another use of the geometric series. EXAMPLE 5. Square wave of period p=2a, f(t)=(-1)^floor(t/a). Verify L(f(t))=(1/s)tanh(as/2). DETAILS. Use the periodic function theorem. The main task is to integrate f(t)exp(-st) on t=0 to t=p. This is done by writing the integral as the sum of two integrals, the first over [0,a] and second over [a,2a]. The integrands simplify and integration is possible, giving integral=(1-exp(-as))^2/s. The rest is a trick, to multiply the fraction for L(f(t)) by exp(as/2)/exp(as/2), distribute, and then discover sinh(as/2) and cosh(as/2) in the fraction. Their quotient is tanh(as/2). EXAMPLE 6. Triangular wave g(t), period p=2a. Verify the identity L(g(t))=tanh(as/2)/s^2. DETAILS. Because g'(t)=f(t) in Example 5, then the parts formula implies sL(g)-g(0)=L(g')=L(f)=tanh(as/2)/s. Evaluate g(0)=0. Then sL(g)=tanh(as/2)/s, which implies the identity. EXAMPLE 7. Forced damped spring-mass system. Let f(t) be the square wave of amplitude 20 and period p=2Pi. Solve the problem x''+4x'+20x=f(t), x(0)=x'(0)=0. ANSWER: Too complicated to write here. See Page 624. DETAILS. Write F(s)=L(f(t)). Then F(s)=20/s+(40/s)(series of (-1)^n exp(-n Pi s), n=1..infinity) Transform the DE into X(s)=F(s)/Z(s), where Z(s)=s^2+4s+20. Expand F/Z into an infinite series and apply the second shifting theorem on each term. EXERCISES 7.5 APPLICATION: Engineering functions This material is used heavily in the maple lab on Laplace theory.