Applied Mathematical Sciences I Volume 15 M Braun Differential Equations and Their Applications An Introduction to Applied Mathematics 2nd Edition Springer-Verlag New York· Heidelberg· Berlin Martin Braun Department of Mathematics Queens College City University of New Y ork Flushing, NY 11367 USA Editors Fritz John Lawrence Sirovich Courant Institute of Mathematical Studies New York University New York, NY 10012 USA Division of Applied Mathematics Brown University Providence, RI 02912 USA Joseph P LaSalle Gerald B Whitham Division of Applied Mathematics Brown University Providence, RI 02912 USA Applied Mathematics Firestone Laboratory California Institute of Technology Pasadena, CA 91125 USA AMS Subject Classifications: 98A20, 98A35, 34-01 Library of Congress Cataloging in Publication Data Braun, Martin, 1941Differential equations and their applications (Applied mathematical sciences ; v 15) Includes index Differential equations I Title 11 Series 5\O'.8s [515'.35] QAI.A647 vol 15 1978 [QA37I] 77-8707 All righ ts reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag © 1975, 1978 by Springer-Verlag, New York Inc Softcover reprint ofthe hardcover 2nd edition 1978 543 ISBN 978-0-387-90266-1 ISBN 978-1-4684-9360-3 (eBook) D0110.1007/978-1-4684-9360-3 To four beautiful people: Zelda Lee Adeena Rachelle, I Nasanayl, and Shulamit Preface This textbook is a unique blend of the theory of differential equations and their exciting application to "real world" problems First, and foremost, it is a rigorous study of ordinary differential equations and can be fully understood by anyone who has completed one year of calculus However, in addition to the traditional applications, it also contains many exciting "real life" problems These applications are completely self contained First, the problem to be solved is outlined clearly, and one or more differential equations are derived as a model for this problem These equations are then solved, and the results are compared with real world data The following applications are covered in this text I In Section 1.3 we prove that the beautiful painting "Disciples of Emmaus" which was bought by the Rembrandt Society of Belgium for $170,000 was a modem forgery In Section 1.5 we derive differential equations which govern the population growth of various species, and compare the results predicted by our models with the known values of the populations In Section 1.6 we derive differential equations which govern the rate at which farmers adopt new innovations Surprisingly, these same differential equations govern the rate at which technological innovations are adopted in such diverse industries as coal, iron and steel, brewing, and railroads In Section 1.7 we try to determine whether tightly sealed drums filled with concentrated waste material will crack upon impact with the ocean floor In this section we also describe several tricks for obtaining information about solutions of a differential equation that cannot be solved explicitly Vll Preface In Section 2.7 we derive a very simple model of the blood glucose regulatory system and obtain a fairly reliable criterion for the diagnosis of diabetes Section 4.5 describes two applications of differential equations to arms races and actual combat In Section 4.5.1 we discuss L F Richardson's theory of the escalation of arms races and fit his model to the arms race which led eventually to World War I This section also provides the reader with a concrete feeling for the concept of stability In Section 4.5.2 we derive two Lanchestrian combat models, and fit one of these models, with astonishing accuracy, to the battle of Iwo Jima in World War II In Section 4.9 we show why the predator portion (sharks, skates, rays, etc.) of all fish caught in the port of Fiume, Italy rose dramatically during the years of World War I The theory we develop here also has a spectacular application to the spraying of insecticides In Section 4.10 we derive the "principle of competitive exclusion," which states, essentially, that no two species can earn their living in an identical manner In Section 4.11 we study a system of differential equations which govern the spread of epidemics in a population This model enables us to prove the famous "threshold theorem of epidemiology," which states that an epidemic will occur only if the number of people susceptible to the disease exceeds a certain threshold value We also compare the predictions of our model with data from an actual plague in Bombay 10 In Section 4.12 we derive a model for the spread of gonorrhea and prove that either this disease dies out, or else the number of people who have gonorrhea will ultimately approach a fixed value This textbook also contains the following important, and often unique features In Section 1.10 we give a complete proof of the existence-uniqueness theorem for solutions of first-order equations Our proof is based on the method of Picard iterates, and can be fully understood by anyone who has completed one year of calculus In Section 1.11 we show how to solve equations by iteration This section has the added advantage of reinforcing the reader's understanding of the proof of the existence-uniqueness theorem Complete Fortran and APL programs are given for every computer example in the text Computer problems appear in Sections 1.13-1.17, which deal with numerical approximations of solutions of differential equations; in Section 1.11, which deals with solving the equations x = J(x) and g(x)=O; and in Section 2.8, where we show how to obtain a powerseries solution of a differential equation even though we cannot explicitly solve the recurrence formula for the coefficients A self-contained introduction to the computing language APL is presented in Appendix C Using this appendix we have been able to teach our students APL in just two lectures viii Preface Modesty aside, Section 2.12 contains an absolutely super and unique treatment of the Dirac delta function We are very proud of this section because it eliminates all the ambiguities which are inherent in the traditional exposition of this topic All the linear algebra pertinent to the study of systems of equations is presented in Sections 3.1-3.7 One advantage of our approach is that the reader gets a concrete feeling for the very important but extremely abstract properties of linear independence, spanning, and dimension Indeed, many linear algebra students sit in on our course to find out what's really going on in their course Differential Equations and their Applications can be used for a one- or two-semester course in ordinary differential equations It is geared to the student who has completed two semesters of calculus Traditionally, most authors present a "suggested syllabus" for their textbook We will not so here, though, since there are already more than twenty different syllabi in use Suffice it to say that this text can be used for a wide variety of courses in ordinary differential equations I greatly appreciate the help of the following people in the preparation of this manuscript: Douglas Reber who wrote the Fortran programs, Eleanor Addison who drew the original figures, and Kate MacDougall, Sandra Spinacci, and Miriam Green who typed portions of this manuscript I am grateful to Walter Kaufmann-Buhler, the mathematics editor at Springer-Verlag, and Elizabeth Kaplan, the production editor, for their extensive assistance and courtesy during the preparation of this manuscript It is a pleasure to work with these true professionals Finally, I am especially grateful to Joseph P LaSalle for the encouragement and help he gave me Thanks again, Joe New York City July, 1976 Martin Braun ix Contents Chapter First-order differential equations 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 Introduction First-order linear differential equations The Van Meegeren art forgeries Separable equations Population models The spread of technological innovations An atomic waste disposal problem The dynamics of tumor growth, mixing problems, and orthogonal trajectories Exact equations, and why we cannot solve very many differential equations The existence-uniqueness theorem; Picard iteration Finding roots of equations by iteration 1.11.1 Newton's method Difference equations, and how to compute the interest due on your student loans Numerical approximations; Euler's method 1.13.1 Error analysis for Euler's method The three term Taylor series method An improved Euler method The Runge-Kutta method What to in practice I 11 20 27 37 43 49 54 64 77 83 87 91 96 102 104 107 III xi Contents Chapter Second-order linear differential equations 2.1 Algebraic properties of solutions 2.2 Linear equations with constant coefficients 2.2.1 Complex roots 2.2.2 Equal roots; reduction of order 2.3 The nonhomogeneous equation 2.4 The method of variation of parameters 2.5 The method of judicious guessing 2.6 Mechanical vibrations 2.6.1 The Tacoma Bridge disaster 2.6.2 Electrical networks 2.7 A model for the detection of diabetes 2.8 Series solutions 2.8.1 Singular points; the method of Frobenius 2.9 The method of Laplace transforms 2.10 Some useful properties of Laplace transforms 2.11 Differential equations with discontinuous right-hand sides 2.12 The Dirac delta function 2.13 The convolution integral 2.14 The method of elimination for systems 2.15 A few words about higher-order equations 121 121 132 135 139 145 147 151 159 167 169 172 179 192 200 209 214 219 227 232 234 Chapter Systems of differential equations 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 Algebraic properties of solutions of linear systems Vector spaces Dimension of a vector space Applications of linear algebra to differential equations The theory of determinants Solutions of simultaneous linear equations Linear transformations The eigenvalue-eigenvector method of finding solutions Complex roots Equal roots Fundamental matrix solutions; eAt The nonhomogeneous equation; variation of parameters Solving systems by Laplace transforms Chapter Qualitative theory of differential equations 4.1 4.2 4.3 4.4 Xll Introduction Stability of linear systems Stability of equilibrium solutions The phase-plane 240 240 249 255 267 273 286 296 309 317 321 331 336 344 348 348 354 361 370 Contents 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 Mathematical theories of war 4.5.1 L F Richardson's theory of conflict 4.5.2 Lanchester's combat models and the battle of Iwo lima Qualitative properties of orbits Phase portraits of linear systems Long time behavior of solutions; the Poincare-Bendixson Theorem Predator-prey problems; or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I The principle of competitive exclusion in population biology The Threshold Theorem of epidemiology A model for the spread of gonorrhea 374 374 381 390 394 404 413 421 428 435 Chapter Separation of variables and Fourier series 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Two point boundary-value problems Introduction to partial differential equations The heat equation; separation of variables Fourier series Even and odd functions Return to the heat equation The wave equation Laplace's equation 446 446 451 453 457 463 468 473 478 Appendix A Some simple facts concerning functions of several variables 484 Appendix B Sequences and series 486 Appendix C Introduction to APL 488 Answers to odd-numbered exercises 498 Index 515 xiii Answers to odd-numbered exercises 11 [COSh V57 + _3_ sinh V57 ]e3t/2 V57 19 y(/)=cos/+%sin/-i/cosl SECTION 2.11 I y(/)=(2+ 3/)e- t + 2H3(/) [ (t- 5)+ (t -1)e -(t-3)] y(t)=3cos2t-sin2t+ HI -cos2t)- Hl-cos2(t-4»H4(t) y(t)=3cost-sint+ itsint+ i H / 2(t) [ (t - !'I1)sint - cost] _I [(7t - 1)+(c O V'Yit I3- S IV'Yit)-t/2 s -ll e 7.y (I) 49 V'Yi (t - 2) -H2 (t)(7t-l)-H 2(t) I3cosV'Yi - [ V'Yi 2(t-2) ] e-(t-2)/2 +V'YiSIll y(t)= tet + Hl(t)[2+ t+ (2t-5)e t - 1] - Hit) [ 1+ t +(2t-7)e t - 2] SECTION 2.12 (a) y(/) = (cosh tt- sinh t/)e 3t / 2- 2H2(t) sinh HI - 2)e3(t-2)/2 y(t)=Hsint-tcost)-H (t)sint y(/)=3te- t + tt 2e- t +3H (t)(t - 3)e-(t-3) SECTION 2.13 e bt _ eat I O b a ttsint 17 a sin at - b sinbt a -b 11 (t - 2)+(t + 2)e- t sin at - at cos at 2a 13 y(t)= t+ ~ sin2t t-sint 15 y(t)= tt y(t)=tsint+(l-~t)e-t SECTION 2.14 I X(/)=cle3t+3c2e4t, y(t)=cle3t+2c2e4t x(t)=2Cl +2c2sint e- 2t , y(t)=c1(I-sint)+c2(cos/+sint)e- 2t x(/)= ~e3t + ~e-t, y(t)= ~e3t - ke- t X(/)=cost e- t, y(t)=(2cost+sint)e- t x(t)=2(tcoS/+3tsint+sint)e t, y(t)= -2tsinte t 505 Answers to odd-numbered exercises 11 x(t)= -4sint- ttsint - toost + 500st In(sect +tant), t sin2 tcost - y(t)= - t sint - tcost - - t sin SECTION 5sintcost + sinzt t + 5(cost -sint)In(sect + tant) 2.15 y(t)=clel+cze-l+c3e21 y(t)=(CI+C2t+c3t2)e2/+c4e-1 y(t)=O l y(t)= - - 2t - tt +(3 - t)e "'(t) = I-cost -lnoost - sint In(sect + tant) 11 "'(1)= _1_ v'2 fI[ sin(t-s)/v'2 cosh(t-s)/v'2 )0 -cos(t - s)/v'2 sinh(t - s)/v'2 ]g(s)dY 13 "'(t) = ~t(e-2/-I)- t sint 15 "'(1)= ~t2[ 17 "'(t) = t-I + tte- - tt)cost+n + kt-1it2)sint] I Chapter SECTION 3.1 7.x=(_~ ~)x, x(3)=(~) XI =X2 X2=X3 X3= -xi x=(~ X3=X4 X4= I-x g o -I -b)x, X(-I)=(~) 13 P~); (b) ( =D; (a) (c) ( I~); (d) ( - t) 15 A=( SECTION 3.2 Yes SECTION No No Yes No 11 Yes 3.3 Linearly dependent Linearly independent (a) Linearly dependent; (b) Linearly dependent (b)YI(t)=e l, Yz(t)=e- I 9,PI(t)=t-I, P2(t)=t -6 1l.xl=(=D 17 XI=YI X2= 506 Vi (Y2- YJ) t -2 !) Answers to odd-numbered exercises SECTION 3.4 I cosV3 t/2 e- / X1(t)= [ (-tcosV3 t/2-tV3 sinV3 t/2)e- '/ 2 I x2(t) = sin V3 t/2 e- / [ (tcosV3 t/2-!sinV3 t/2)e-'/2 X1(t)=( SECTION 0), x 2(t) e' 5.0 -97 Yes No 11 (b) No 25) 10 ,BA= C55 12 10 11 No solutions 3.6 AB=( ~ 10 AB=(i I(-3 18 13 -i ( -0~ ~ SECTION Yes 2te' 3.5 -48 SECTION = ( e' ) I I D,BA=O -4 -12 14 -2; -1 9) 18 -18 52 -I 18 I I n D 2; -i ( 12; -1 1-2; 11 -2;) -1+; 1+; 0) 1- ; -1-; 3.7 No solutions x = ( ~ ) 11.A=I,-1 13 (a)A=-I; (b)x=i( -D+c( =D 507 Answers to odd-numbered exercises SECTION 3.8 31 41 X(t)=ct(De +c2(De X(t)=c t ( _ne-'+C2( _ne-'+c3Gle81 SECTION 3.9 X(t)=[C ( t )+c ( 2sin~ )]e- 21 l-smt cost+smt -~l3 +C2(C?~2tl +C3( s~2t lj sm2t -cos2t X(t)=el[Ct( x(t)= SECfION (-i2) e- 21 + [-Vi Vi Vi t] sin t-Vi cos cosV2 t-V2 sinV2 t -3cosV2 I cost 2cost+smt x(O) = (~J 3.10 x(t)-c,(!je-"+U)+c,( D x(t)-C,me-"+(i)+C'( -D f)e (I-I) je-' je-' S X(I) = ( l 13 (b) I~ [ 2Ie ' - 5e -16e14e ' + lOe21 -24e- 15 1+ A (eat -I) a 2/ e 21 ~ x(t)= 7et- 5e 2t + 8e- t 508 S x(t) = ( - e t + 5e 2t - 4e- t 3e t - 5e 21 -2e- / ] -3e + 5e 21 + 8e- ge ' - 5e 2/ -4e- -2e ' -lOe 21 +12e- 6e ' +lOe2' -6e- t )e- I Answers to odd-numbered exercises SECTION I [e-21+5e21-e31 5e 21 -5e 31 5e 31 - 5e 21 + 5e 31 2/ I [6e -cost - 2sint - 2e 21 + 2cost + 2sint - e21 +6cost+2sint -lOsint t 3" 3.11 -e- 21 +e31 4e - 21 - 5e 21 + e 31 3" 3e21 -3cost-sint 5sint [(1+ I)e-' -te-I te- I I A=U 3 SECTION - -"] (t+ l)e- I - e- 21 -te- l +e- 21 te- I e -e e- 21 -e- e- I A=n C~ -I) -I -I I 21 2e -2cost+sintj e 21 -cost+3sint 5cost -25 -6 13 30) -24 26 ll.No 3.12 t X(t)=2e ' ( tcost + 3ts~nt +sint) -tsmt -4sint - ttsint - tcost + cos t In(sect +tant) x(t)= t sin2 tcost 5sintcost + 5sin2 t - t sin t - tsint - tcost + sin t cos t - + 5(cost - sint) In(sect + tant) SECTION 3.13 / x(t)=e- (D+e 4/ ( -n 2/ x(t)= - t(De- ' + i(De + i(?)-t( D+3( Del X(t)=2e ' ( tcost+ 3ts~t +sint) -tsmt x(t)= COs2t.+sin2t), 2sm2t cos2t ) {( sin2t+cos2t ' ( ll x(t)= ( I-t) 21 ~ e 509 Answers to odd-numbered exercises t - t 2- tt -tt + -fit 13 X(t)= -tt t + it Chapter SECTION 4.1 x=O,y=O; x=O,y=l; x=l,y=O; x=t,y=~ x=Oy=O , , z=O', x=E d' y=!! b' z=_(£+a d b2 ) x=O,y=Yo; Yo arb x=O,y= -2, z=Tl'TT x = xo, y = I; Xo arb X=Xo, y= -I; Xo arb SECTION 4.2 Stable Unstable S Asymptotically stable Asymptotically stable 11 x(t)= I is stable; SECTION Stable x(t)=O is unstable 4.3 x=O, y =0 is unstable; x= I, y =0 is unstable; x= -I, y =0 is unstable; x=O, y=2 1/ is stable; x=O,y= -2 1/ is stable x = 0, y = I is unstable; x= l,y=O is unstable; x = 0, y = - I is unstable; x= -1,y=O is stable S x = 0, y = mr is unstable for all integers n Unstable Unstable 15 Unstable 17 Unstable SECTION 11 Stable 13 Stable 4.4 y=cos(x-I)2 y=tanx The points X=Xo,y=Yo with xo+Yo= -I, and the circles x 2+y2=e 2, minus these points The points x=O,y=Yo; y = ee -(2/3)e'x, x < O x=xo,y=O, and the curvesy=ee-(2/3)e''', x>O; 11 The points x = 0, y = Yo, and the curves (ad-be) (by-dx)d lnle-~I=k, x>O (ad-be) d lnlc-~I=k, xec and is stable =0, X2=0,y=0; Xl =c/ d, x2=nc/[d(al +a:J],y=na2-ar-ala2; assuming that na2> a; + ala2' I+al l-a2 5.(b)(I)Yl=l+ ,Y2=1 ; (1l)Yl=I'Y2=0; (c)a2=~ Xl ala2 SECTION -ala2 4.11 (a) rI+AI 2/2=ylnS-rS+c; (b) Yes (a) 0.24705; (b) 0.356; (c) 0.45212; (d) 0.60025; (e) 0.74305; (f) 0.77661 SECTION 4.12 Chapter SECTION An= 5.1 li'IT2 (2n+ 4/2 A=O,y=C; ,y(x)=csm -n 2'IT2 (2n+ 1)'lTx 21 n'ITx I A= -12 -, y(x)=ccos y(x)= csinh vi ->.0 x, where sinh vi -AO 'IT = vi ->.0 y(x)=csin V>:;, x, where tan V>:;, 'IT = V>:;, A= -1, y(x)= cex; A= n 2, y(x)= c[ncosnx +sinnx] cosh vi ->.0 'IT; 511 Answers to odd-numbered exercises SECTION 5.3 I u(x,t) = sin t7Txe-1.71w1t/4 + sin i7Txe-(1.71)2Sw2t/4 u(x, t) = sin 27TX e(1-4,,2)t -7 Sin47TX e(I-I6.r2)t S u(t,y)=e2(t+Y )+e- 3(t+ y) u(t,y) = e- Ste-4y +2e- 7te- 6y -14e13te14y (a)X"-I'X=O; Y"-(I'+A)Y=O; T'-M 2T=O; (b) u(x,y,t) = sin ,:X sin "'; ea2n2.il(6L~t/a262 SECTION I f(x)= 5.4 ~ [ sinrx + Sinj7TX + Sing7TX + ] [ Sin7TX _ Sin27TX + Sin37TX + ] • f( X ) = ~ 71' _ £.J ~ -1 [ rnr n7TX ( l-cosrnr) sm rnrx] S f (x)= sm-cos-+ 71' n= I n 2 2 l e ( -1)"-1 [ rnrx rnrx] 2 lcos-I rnrsm-I n= I I + n 71' el 00 I(e l ( -1)"-1) n7TX f(x) = T +2 ~ 2 cos-I n-I I +n 71' f(x) = e l -l 21 + 00 ~ el-l ~(-I)"(el-e-/)[ rnrx rnrx] ll f(x) = -1- + £.J 2 lcos +rnrsm-I I n=1 I +n 71' 71'2 13.f(x)=*sinx-t sin 3x 00 17.(a)f(x)=T+4~ n=1 SECTION (-I)"cosnx n 5.5 e-l ~ 2(I-cosrnr/e) I f(x)= - - + £.J cosrnrx en_I l+n 27T2 _ 3a f(x)- -4 + ~ 4a(cosrnr/2 -1) £.J n-I 2 n 71' rnrx cos2a ~ - [ cos rnr /2 -1- (-1) n] cosrnrx S f(x)= -I + -21 £.J 71'2 n= I n I 7.f(x)=!: ~ !(cosrnr/2 -(-1)")sin rnrx 71' n=1 n f(x)=sin2x 4[cos2x cos4x + ] ; O[...]... solution of ( 9) is a solution of ( 3) and vice-versa .) Thus, can we choose p.(t) so that p.(t)(dy I dt) + aCt) p.(t)y is the derivative of some simple expression? The answer to this question is yes, and is obtained by observing that d dy dp dt p.(t)y=p.(t) dt + dt y Hence, p.(t)(dyldt)+a(t)p.(t)y will be equal to the derivative of p.(t)y if and only if dp.(t)1 dt = aCt) p.(t) But this is a first-order linear... the form d dtln1y(t)l= -aCt) ( 3) We then found Inly(t)l, and consequently y(t), by integrating both sides of ( 3) In an exactly analogous manner, we can solve the more general differential equation ( 4) where f and g are continuous functions of y and t This equation, and any 20 1.4 Separable equations other equation which can be put into this form, is said to be separable To solve ( 4), we first mUltiply... mUltiply both sides by fey) to obtain the equivalent equation dy ( 5) fey) dt =g(t) Then, we observe that ( 5) can be written in the form d dt F(y(t ) = get) where F(y) is any anti-derivative of f(y); i.e., F(y) = quently, F(y(t )= Jg(t)dt+c ( 6) Jf(y)dy Conse- where c is an arbitrary constant of integration, and we solve for y from ( 7) to find the general solution of ( 4) ( 7) =y(t) Example 1 Find the general... linear homogeneous equation for p.(t), i.e (dp.1 dt) - aCt) p = 0 which we already know how to solve, and since we only need one such function p.(t) we set the constant c in ( 7) equal to one and take p.(t)=exp(f a(t)dt) For this p.(t), Equation ( 9) can be written as d dt p.(t)y = p.(/)b(t) (1 0) To obtain the general solution of the nonhomogeneous equation ( 3), that is, to find all solutions of the nonhomogeneous... integral (anti-derivative) of both sides of (1 0) which yields p.(t)y = f p.(t)b( t)dt+ c or y= p.~t)(f P.(/)b(/)dt+c)=ex p( - fa(t)dt)(f P.(t)b(t)dt+C) (1 1) Alternately, if we are interested in the specific solution of ( 3) satisfying the initial condition y (to) = Yo, that is, if we want to solve the initial-value problem dy dt +a(t)y=b(t), then we can take the definite integral of both sides of (1 0) between... of ( 5) to obtain that Inly(t)l= - f a(t)dt+c I where C I is an arbitrary constant of integration Taking exponentials of both sides yields ly(t)l=ex p ( - fa(t)dt+cl)=cex p ( - fa(t)dt) or Iy(t)exp(f a(t)dt)1 = c ( 6) Now, y(t) exp(f a(t)dt) is a continuous function of time and Equation ( 6) states that its absolute value is constant But if the absolute value of a continuous function g(t) is constant then... integrate both sides of ( 6) between to and t to obtain that F{y(t )- F(yo)= fl g(s)ds ( 8) 10 If we now observe that F(y)-F(yo)= fYf(r)dr, ( 9) Yo 21 I First-order differential equations then we can rewrite ( 8) in the simpler form fY j(r)dr= fl g(s)ds Yo (1 0) 10 Example 3 Find the solution y (t) of the initial-value problem dy 3 eY dt - ( t + t ) = 0, ( y I) = 1 Solution Method (i) From Example 2, we know that... sides of the equation Remark 4 An alternative way of solving the initial-value problem (dy / dt) + a(t)y = bet), y (to) = Yo is t