1、Section 7.5Solution of Higher Order Homogeneous LDE with Constant Coefficients12,a awhereare all constants.The general form of the second order homogeneous LDEwith constant coefficients is120,(1)ya ya yis called a linear differential()(1)1(),nnnya ya yf x The equationequation with constant coefficie
2、nts or linear equation with constant coefficients.,(1,)iain whereare all constants,2Solution of Higher Order Homogeneous LDE with Constant Coefficientsthere must have 0,xe The solution of characteristic equation is called eigenvalues or 2120.aaCharacteristic equation(特征方程特征方程)Since characteristic ro
3、ots(特征根特征根).120ya ya y3into the differential equation,we have If we substitute xye 2,.xxyeye 2120.xeaa HenceSolution of secondr Order Homogeneous LDE with Constant Coefficients.xye Obviously,for any characteristic root,there must have a solution We will show the solution of the second order LDE with
4、 respect to the characteristic roots in three cases.120ya ya y42120.aaCharacteristic equation12R12R1,2iThe Solution of Second Order LDE with Constant Coefficients:Case IIf there have two different real roots,1 and 2,of the characteristicequation,then we have22.xye and11xye 221210(10)ya yyaaa 5121)Ra
5、re two solutions of the equation(1),andso are linearly independent.2121()(),.()xyxeCxIy x 12,yy12,yyThe Solution of Second Order LDE with Constant Coefficients:Case ITherefore,all the solution of the second order LDE(1)can be expressed as are all constants.2cwhere 1cand1212,xxyc ec e6221210(10)ya yy
6、aaa 121)RThe Solution of Second Order LDE with Constant Coefficients:Case IExample Find the solution of the equation40.yySolution The characteristic equation of this equation is240.Then,we have24.and10 Therefore,the solution can be expressed as412.xyc ec Finish.7The Solution of Second Order LDE with
7、 Constant Coefficients:Case II8221210(10)ya yyaaa 122)we can only find one particular solution112,2a If there have two repeated real roots,11.xye How to find another solution y2 hat is linearly independent of y1?2211()(),()()()yxyxCxIAssumeh xy xy x The Solution of Second Order LDE with Constant Coe
8、fficients:Case II9221210(10)ya yyaaa 122)12211Assumei.e.()(),()()()().()xyxh xyxh x y xh x ey xSubstituting y2 into the equation(1),we have10 or 0()().xh x eh x21112110&20aaaIt is easy to get the general solution of the above equation is().h xcxc121()()()().xyxh x y xcxc eThe Solution of Second Orde
9、r LDE with Constant Coefficients:Case II10221210(10)ya yyaaa 122)21112110&20aaa12()().xyxcxc e11;xye The general solution of the LDE(1)is 1111212xxxxxyc ececce112()()().xyAy xByxBcxBcA e1c2cThe Solution of Second Order LDE with Constant Coefficients:Case II 12.xycc x e Example Find the solutions of
10、the equation20.yyySolution then the solution121,The characteristic equation of this equation isof this equation isFinish.112210.We obtain the roots areThe Solution of Second Order LDE with Constant Coefficients:Case III3)If the characteristic equation has a pair of conjugate complex roots Then,we kn
11、ow that the homogeneous equation has two particular solutions 2.iand1iand 2.ixye 1ixye 12221210(10)ya yyaaa The Solution of Second Order LDE with Constant Coefficients:Case IIIWe can get the real and imaginary parts of y1 and y2 by 121sin2xyyexi 2cossin.xyexix By Euler formulae,we can rewrite this t
12、wo solutions as 1cossinxyexix and 121cos2xyyex and13221210(10)ya yyaaa The Solution of Second Order LDE with Constant Coefficients:Case IIIThen,we can express all the solution as 12cossin.xyecxcx 14221210(10)ya yyaaa of the equation(1),and they are linearly independent.Therefore sinxex cosxex andare
13、 also the solutionsThe Solution of Second Order Differential Equation with Constant Coefficients:Case IIIExample Find the solution of the equation 220.xxxSolution The characteristic equation of this equation is 2220.Therefore,the solution is 121,1.ii Then,12cossin.tectct 112cossintxectctExample Find
14、 the solution of the equation 0.yy Solution The characteristic equation of this equation is 210.Therefore,the solution is 12,.ii Then,12cossin.cxcx 012cossinxyecxcxFinish.Finish.15Solution of Higher Order Homogeneous LDE with Constant CoefficientsTo the equation()(1)10,nnnya ya y we have its charact
15、eristic equation:the solution must have the term11cos,cos,cos,sin,sin,sin;xxkxxxkxex xexxexex xexxex is a single root of the characteristic equation,the solution 1 1)If is a multiple complex root of orderthen1i3)If,(2),k k must have the term1;xe must have the term1111,;xxxkexexe then the solution1 2
16、)If,(2),k k is a multiple root of order16()(1)10.nnnaa ThenSolution of Higher Order Homogeneous LDE with Constant CoefficientsExample Find the solution of the equation 230.xxxSolution The characteristic equation of this equation is 32230.Therefore,the solution is 1230,1,3.Then,3123.ttxcc ec e Exampl
17、e Find the solution of the equation 0.yy Solution The characteristic equation of this equation is 310.Therefore,the solution is 12,3131,.2i Then,22333cossin.22xecxcx 1xyc e Finish.Finish.17Solution of Higher Order Nonhomogeneous LDE with Constant CoefficientsTheorem Let y be any particular solution
18、of an nth-ordernonhomogeneous linear differential equation.yYy be the general solution of the 1122nnYc yc yc yandThen,the general solution of the corresponding homogeneous equation.nonhomogeneous equation is()(1)11()()()()nnnnyP x yPx yP x yF x This theorem says that to find the solution of nonhomog
19、eneous linear(2)Find a particular solution of nonhomogeneous linear equation.(1)Find the homogeneous solutions of homogeneous linear equation;Then we combine them together to form the solution.differential equations with constant coefficients can follow two steps:18Solution of Higher Order Nonhomoge
20、neous LDE with Constant Coefficients0.yyySolution The corresponding homogeneous equation is Example Find the solution of.xyyyeand the eigenvalue are210The characteristic equation is1,215,2 then the general solution of the homogeneous equation is15152212.xxxyc ec eeis a solution of the xye Moreover,i
21、t is easy to see that Therefore,the general solution is nonhomogeneous equation.15152212.xxYc ec eFinish.19Solution of Higher Order Nonhomogeneous LDE with Constant CoefficientsLet us begin from the second order nonhomogeneous LDE12().ya ya yF x,where is a constant and()()xF xx e (1)If(0).m 1110(),m
22、mmmxb xbxb xb 202()()sinxF xx ex or(2)If 1()()cosxF xx ex ,where and are1110(),mmmmxb xbxb xb constants and 0.m where Solution of Higher Order Nonhomogeneous LDE with Constant CoefficientsWe can expect the particular solution of the equation as is a polynomial is determined by the equation.()Z xwher
23、e*()(),xyxZ x e ,where is a constant and()()xF xx e (1)If(0).m 1110(),mmmmxb xbxb xb 21thenis a particular solution of the equation,If assume *()()xyxZ x e 12()ya ya yF x*()()()xxyxZ x eeZ x *2()()2()()xxxyxZx eeZ xeZ x Solution of Higher Order Nonhomogeneous LDE with Constant Coefficients2121()()(2
24、)()()().aa Z xa Z xZxx,where is a constant and()()xF xx e (1)If(0).m 1110(),mmmmxb xbxb xb 22Then,12()ya ya yF xSubstituting them into the equation,we have212()2()()()()()()xxxxxxxZx eZ x eZ x eaZ x eZ x ea Z x ex e 1110(),mmmmxb xbxb xb mth polynomial1mmxmx21()mmxm mx?Solution of Higher Order Nonho
25、mogeneous LDE with Constant Coefficients2121()()(2)()()()aaZ xa Z xZxx2120,aaSince By comparing the coefficients,we can determine,(0,).miBiB(a)If is not an eigenvalue of the characteristic equation.can be assumed as()Z x1110(),mmmmZ xB xBxB xB 231110(),mmmmxb xbxb xb Solution of Higher Order Nonhomo
26、geneous LDE with Constant Coefficients2121()()(2)()()()aaZ xa Z xZxx2411101110(),(),mmmmmmmmxb xbxb xbZ xB xBxB xB 11101211()(1)mmmmmmmmZ xB xBxB xBmB xmBxB 121212312()(1)2(1)(1)(2)2mmmmmmmmZxmB xmBxB xBm mB xmmBxB ()()mQxx compare the coefficients,(0,).miBiBSolution of Higher Order Nonhomogeneous L
27、DE with Constant Coefficients(b)If is a single eigenvalue of the characteristic equation.21210but20.aaa We have()Z x 1110().mmmmZ xx B xBxB xB isThen we can assume that 252121()()(2)()()()aaZ xa Z xZxxBy comparing the coefficients,we can determine,(0,).miBiB 1(2)()()()a Z xZxx (c)If is a repeated ei
28、genvalue of the characteristic equation.()Z x21210 and20.aaa We have isThen we can assume that 21110().mmmmZ xxB xBxB xB Solution of Higher Order Nonhomogeneous LDE with Constant Coefficients262121()()(2)()()()aaZ xa Z xZxxBy comparing the coefficients,we can determine,(0,).miBiB()()Zxx ,where is a
29、constant and()()xF xx e (1)If(0),m 1110()mmmmxb xbxb xb while121110111021110221212121;(00&200&2);.;0mmmmmmmmmmmmB xBxB xBZ xx B xBxB xBxB xaaaaaaaBaBxxB Summing-upthe particular solution can be assume as*()xyZ x e whereSolution of Higher Order Nonhomogeneous LDE with Constant Coefficients27Solution
30、of Higher Order Nonhomogeneous LDE with Constant CoefficientsSolution Example Find the general solution of the equation 256.xyyyxeThe characteristic equation of the corresponding homogenousequationis560yyy2560.The eigenvalues of the characteristic equation are 122,3.Then 2312.xxYc ec ethe general so
31、lution of the homogeneous equation is Since =2 is a simple eigenvalue,we assume that the particularsolution for the nonhomogeneous equation is *201().xyx BB x e28Solution of Higher Order Nonhomogeneous LDE with Constant CoefficientsSolution(continued)29Example Find the general solution of the equati
32、on 256.xyyyxeWe substitute them into the nonhomogeneous equation,then11022.B xBBx*201().xyx BB x e *2201102210102()(2)22().xxxyx BB x eB xB eB xBB xBe It is easy to get *221100184(2)42xyB xBBxBBe Solution of Higher Order Nonhomogeneous LDE with Constant CoefficientsSolution(continued)Comparing coeff
33、icients of the powers with the same degree on bothHence,the desired particular solution is 11022.B xBBxsides we have101,1.2BB 2*2,2xxyxe andthe general solution of the given equation is 223212.2xxxxyc ec exeFinish.30Example Find the general solution of the equation 256.xyyyxe 212.xYecc xSolution The
34、 corresponding homogeneous equation is 23()4()4()1.Z xZ xZ xxxx,therefore,the general2 This equation has a repeated eigenvalue 440.yyysolution of homogeneous equation iswe assume that the then we substitute*320123(),yZ xb xb xb xbparticular solution is it into the nonhomogeneous equationExample Find
35、 the solutions of 23441.yyyxxxSolution of Higher Order Nonhomogeneous LDE with Constant Coefficientsis0 Since not an eigenvalue of the characteristic equation,31By rearranging the terms,we have 323201021032144124864421,b xbbxbbbxbbbxxxthen01021032141,4121,4861,4421.bbbbbbbbbThis gives that012311513,
36、1,.488bbbbTherefore,the particular solution of nonhomogeneous equation is*3211513()488yZ xxxxand the general solution is 2321211513.488xyecc xxxxSolution of Higher Order Nonhomogeneous LDE with Constant CoefficientsFinish.322440,Solution The characteristic equation of the corresponding homogeneousth
37、en it has a repeated eigenvalue Example Find the solutions of where a is a real constant.44,xyyye equation is 1,22.Therefore,the general solution 212.xYcc x e of the homogeneous is Suppose that the particular solution of the nonhomogeneous equation is*0220,2,2.xxb eyx b e Solution of Higher Order No
38、nhomogeneous LDE with Constant Coefficients33Solution of Higher Order Nonhomogeneous LDE with Constant CoefficientsTherefore,the general solution of the nonhomogeneous equation is 2122222121,2,21,2.2xxxxcc x eeycc x ex e 201,2,21,2.2b Solution(continued)By substituting,we obtain that 0*220,2,2.xxb e
39、yx b e Finish.34Solution of Higher Order Nonhomogeneous LDE with Constant CoefficientsIn this case,the equation can be seen as the real or imaginary part of differential equation with a complex constant as 12(),ixya ya yex We can expect that the particularare real constants.2awhere 1aandsolution sho
40、uld be in complex form as()().RIyyxiyx2()()sinxF xx ex or(2)If 1()()cosxF xx ex ,where and are1110(),mmmmxb xbxb xb constants and 0.m where 35Solution of Higher Order Nonhomogeneous LDE with Constant Coefficientswill be the solution of()IyxIt is easy to see that()Ryxandand12()sin,xya ya yext 12()cos
41、xya ya yexx respectively.By this way,the problems has been changed into find the solutionand this is the problem we have discussed in(1),except the constant 12(),ixya ya yex of nonhomogeneous linear differential equation of i here are a complex constant.Then we can use the same wayin(1)to find the s
42、olution in(2).36Solution of Higher Order Nonhomogeneous LDE with Constant CoefficientsExample Find the general solution of the equation cos.xyyxex Solution 3712()sin,xya ya yext 12()cosxya ya yexx 12()ixya ya yex ()cos,xF xxex Since we have and 1().iixxNow,we try to solve the equation 1 i xyyxe Solu
43、tion of Higher Order Nonhomogeneous LDE with Constant Coefficients38 1 i xyyxe 1)Find the general solution of the eq uation 120.ya yyYaThe characteristic equation of equation is 0yy 210.12,.ii Then,12cossin.cxcx 012cossinxYecxcxTherefore,the general solution of is 0yy 1iiSolution of Higher Order Non
44、homogeneous LDE with Constant Coefficients39 1 i xyyxe 2)Assume the particular solution is*()().ixyZ x e Since the complex number is not an eigenvalue of the characteristic equation,that is,212()()0,iaia1i 10().Z xB xBwe can assume that3)Substitute into the equation*y()12().ixya ya yx e or 3)Substit
45、ute into the equation()Z x 2121()()()()2()()()aaZ xaixiZiZxx 1iiSolution of Higher Order Nonhomogeneous LDE with Constant Coefficients40 1 i xyyxe 1ii2121()()()()2()()()aaZ xaixiZiZxx 1012();0;1;1Z xB xB aaii101(12)()(22).iB xBi Bx101(12)1.(12)2(1)0i Bi Bi B 120112(12)5.2(1)(12)2142525iBiiiiB 4)By c
46、omparing the coefficients,determine,(0,).miBiB Solution of Higher Order Nonhomogeneous LDE with Constant Coefficients41 1 i xyyxe 112214*.525i xiiyxe 5)The solution of is 12()cosxya ya yx ex where *,(*).RRyYyyR y 12214*cossin52522142142cossincossin525525525525xxiiyxxix exxxxexxixix Solution of Highe
47、r Order Nonhomogeneous LDE with Constant Coefficients42Therefore,the general solution of the nonhomogeneous equation iscosxyyxex*122214cossincossin.525525RxyYyxxcxcxexx2214*(*)cossin525525xRxxyR yexxFinish.Example Find the general solution of the equation cos.xyyxex Solution of Higher Order Nonhomog
48、eneous LDE with Constant CoefficientsThe Method of Undetermined CoefficientsWe want to find a particular solution of the equation *(),ixkyx eZ x We may assume the solution is here k=0 if i are not eigenvalues;k=1 if i are eigenvalues.Then we substitute it into the nonhomogeneous equation,by themetho
49、d of comparing coefficients we can determine the complexpolynomial Z(x)whose degree is the same as the degree of ().x 43 12().ixya ya yex ImaginarySolution of Higher Order Nonhomogeneous LDE with Constant CoefficientsThus,substituting Z(x)=R(x)+iI(x)into the equation of y*:we have44 *(),ixkyx eZ x Z
50、(x)is a complex polynomial.Assume that Z(x)=R(x)+i I(x),where R(x)and I(x)both are real polynomials.*()()()()cossin()cos()cos()sin()sinixkkxixkyx eR xiI xx eR xiI xxixx eR xxiI xxiR xxI xx RealSolution of Higher Order Nonhomogeneous LDE with Constant CoefficientsWe find*()cos()sin,kxRyx eR xxI xx*()
