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Chapter 8: Q-14E (page 434)

Question:In Problem find the first three nonzero terms in the power series expansion for the product f(x)g(x).

Short Answer

Expert verified

The required product is, f(x).g(x)=1.

Step by step solution

01

Given power series expansion

The series expansion of the given series is given by

02

Find the product of the power series expansion

Multiplying both the series

There is only one non-zero term in the series expansion, that is f(x).g(x)=1.

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Most popular questions from this chapter

In Problems 13-19,find at least the first four nonzero terms in a power series expansion of the solution to the given initial value problem.

y''-e2xy'+(cosx)y=0y(0)=-1,y'(0)=1

For Duffing's equation given in Problem 13, the behaviour of the solutions changes as rchanges sign. Whenr>0, the restoring forceky+ry3becomes stronger than for the linear spring(r=0). Such a spring is called hard. Whenr<0, the restoring force becomes weaker than the linear spring and the spring is called soft. Pendulums act like soft springs.

(a) Redo Problem 13 withr=-1. Notice that for the initial conditions,y(0)=0,y'(0)=1 the soft and hard springs appear to respond in the same way forsmall.

(b) Keepingk=A=1and,蝇=10 change the initial conditions toy(0)=1andy'(0)=0. Now redo Problem 13 withr=卤1.

(c) Based on the results of part (b), is there a difference between the behavior of soft and hard springs forsmall? Describe.

The equation

(1-x2)y"-2xy'+n(n+1)y=0

where nis an unspecified parameter is called Legendre鈥檚 equation. This equation appears in applications of differential equations to engineering systems in spherical coordinates.

(a) Find a power series expansion about x=0 for a solution to Legendre鈥檚 equation.

(b) Show that fora non negative integer there exists an nthdegree polynomial that is a solution to Legendre鈥檚 equation. These polynomials upto a constant multiples are called Legendre polynomials.

(c) Determine the first three Legendre polynomials (upto a constant multiple).

Find at least the first four nonzero terms in a power series expansion about x0for a general solution to the given differential equation with the given value for x0,

y''+(3x-1)y'-y=0;x0=-1

Use Table 6.4.1 to find the first three positive eigen values and corresponding eigen functions of the boundary-value problem\(xy'' + y' + \lambda xy = 0,y(x),y'(x)\)bounded as \(x \to {0^ + },y(2) = 0\). (Hint: By identifying \(\lambda = {\alpha ^2}\), the DE is the parametric Bessel equation of order zero.)
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