1、Section 7.61General Solution of Higher Order Nonhomogeneous Linear Differential Equations with Variable CoefficientsThe problem to solve a higher order nonhomogeneous lineardifferential equations with variable coefficients is much difficultthan that with constants coefficients.Eulers differential eq
2、uationThe general form of the Eulers differential equation is 11111(),nnnnnnnnd xdxdxta tata xf tdtdtdt are all constants.(1,)ia in where To solve this equation,let ln,t orte then Eulers equation can be changed into differential equation with constants coefficients.2General Solution of Higher Order
3、Nonhomogeneous Linear Differential Equations with Variable CoefficientsExample Find the general solution of the equation 20.t xtxxSolution This is an Euler differential equation.Let ln,t orte so that 222221,1.dxdx ddxdtddtt dd xd xdxddttd 3General Solution of Higher Order Nonhomogeneous Linear Diffe
4、rential Equations with Variable CoefficientsExample Find the solution of 2235ln.x yxyyxxSolution This equation is an Eulers equation,then we let ln.tx ortxe Then dydtydtdx 1,tyx 211ttdtyyyxxdx21().ttyyxSubstitute these back to the original equation,we have 245.tttyyyte4General Solution of Higher Ord
5、er Nonhomogeneous Linear Differential Equations with Variable CoefficientsSolution(continued)The corresponding homogeneous equation is 450.ttyyyIts characteristic equation is2450,and its eigenvalue are 125,1.Therefore,the general solution of the homogeneous equation is512.ttyc ec e 245tttyyyte5Gener
6、al Solution of Higher Order Nonhomogeneous Linear Differential Equations with Variable CoefficientsSolution(continued)Assume that the particular solution of the nonhomogeneous equation is*2(),tyatb ewe have*2()(444).tyeatab and*2()(22)tyeatab Substitute them back into the differential equation with
7、constantsTherefore,we have1,0,9ab coefficients,we have*21.9tyte then 521219tttyc ec ete 52211ln.9cc xxxx245tttyyyte6ReviewStructure of Solutions of Linear Differential EquationsSolution of Higher Order Homogeneous Linear Differential Equations with Constant CoefficientsSolution of Higher Order Nonho
8、mogeneous Linear Differential Equations with Constant Coefficients*Solution of Eulers Differential Equation7Section 7.78Mathematical ModelsMathematical Model is an idealization of the real-world phenomenon and never a completely accurate representation.9Real-world dataModelPredications/explanationsM
9、athematical conclusionsSimplificationAnalysisInterpretationVerificationSome Applications for Differential Equations10In general,the procedures for applying differential equations to solve practical problems are the following:(1)Establish the approximate differential equation and initial conditions u
10、sing knowledge of mathematics and related sciences;(2)Find the general solution of the equation and then determine the desired particular solution using the initial conditions.Example Find the equation of the curve such that the distancebetween any point P on the curve and the origin is equal to the
11、 distance between the point P and the point Q which is an intersection point of the tangent of the curve at point P and the x-axis.Some Applications for Differential Equations11O(,0)Q X(,)P x yxySolution Suppose that P(x,y)is anypoint on the desired curve y=y(x).By theassumption we know that the con
12、ditiondetermining the curve is.OPPQ To find the length|PQ|we first write downthe equation of the tangent PQ as follows:(),Yyy Xx where(X,Y)is the variable point on the tangent.Let Y=0.The abscissa of the point Q is,yXxy so that2222|().yPQxXyyy Some Applications for Differential Equations12Thus,the c
13、oordinate representations of the condition is2222.yxyyy The general solution is easily obtained as follows:yCx or.Cyx Obviously,there are two families of curves both satisfying the requirement.One is a family of hyperbolas;Cyx the other is a family of rays y=Cx.Finish.Some Applications for Different
14、ial Equations13When an atom emits some of its radioactive mass,the remainder of the atom becomes an atom of some new element.This process of radiation and change is radioactive decay.Radioactive carbon-14 decays into nitrogen.Radium,through a number of intervening radioactive steps,decays into lead.
15、Experiments have shown that at any given time,the rate at which a radioactive element decays is approximately proportional to the number of radioactive nuclei present.Thus,the decay of a radioactive element is described by the equation dx/dt=-kx,k 0.If x(0)=x0,the number left at any later time t wil
16、l be0(),0.ktx tx ek Some Applications for Differential Equations14The half-life of a radioactive element is the time required for half of the Radioactive nuclei present in a sample to decay.The next example show the surprising fact that the half-life is a constant that depends only on the radioactiv
17、e substance and not on the number of radioactive nuclei present in the sample.ExampleFinding kSuppose that the half-life of a radioactive substance is T.Then we have the following equation001.2kTx ex Some Applications for Differential Equations15Solve Algebraically121ln211ln2ln2kTekTkTT Take ln of b
18、oth sides.Divide by y01lnlnaaFinish.Scientists who do carbon-14 dating use 5568 years for its half-life.We have k=(ln 2)/5568.Some Applications for Differential Equations16Example Using Carbon-14 to Date000()1()lnln.ln2()ktx tx exx tTtkxx t 0How to find?()xx t(0)(0)Since()(),(0)(0),we have.()()xxx tkx txkxx tx t
