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Chapter 8: Q-32E (page 435)

Question: In Problems 29鈥�34, determine the Taylor series about the point X0for the given functions and values of X0.

32. f(x)=ln(1+x), x0 =0

Short Answer

Expert verified

The required expression is鈭�n-1鈭�(-1)n=1n.(x)n .

Step by step solution

01

Taylor series

For a function f(x) the Taylor series expansion about a point x0 is given by,

f(x-x0)=f(x0)+f'(x0).(x-x0)+f''(x0).(x-x0)22!+f'''(x0).(x-x0)33!+...

02

Derivatives of function at x0

We have to calculate the Taylor series expansion for,f (x)= ln (1 + x) at.

Calculating the derivatives of function at x0,

f (x) =ln (1+x) then f (x0)=0

f'(x) = 11+xthenf '(x0)=1

f''(x) = -1(1+x)2 then f''(x0)= -1

f'''(x) = 21+x3then f'''(x0) = 2

f''''(x) = -6(1+x)4 then f''''(x0) = -6

03

Substitute the derivatives in Taylor series

Substituting the above derivatives in Taylor series expansion for the function atx0=0, then,

ln(1+x)=0+1.(x-0)-1.(x-022!+2x-033!-6.(x-0)44!+....

= x-x22+x33-x44+...

= role="math" localid="1664200514125" 鈭�n-1鈭�(-1)n+1n.(x)n

Hence, the required expression is鈭�n-1鈭�(-1)n+1n.(x)n

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Most popular questions from this chapter

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).

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four nonzero terms in a power series expansion about鈥檚 x=0 of a general solution to the given differential equation.

(1-x2) y"-y'+y=tan x

Question: In Problems 1鈥�10, determine all the singular points of the given differential equation.

1. (x+1)y"-x2y'+3y = 0

In Problems 1-10, use a substitution y=xr to find the general solution to the given equation for x>0.

x2y"+xy'(x)+17y=0

Aging spring without damping. In a mass-spring system of aging spring discussed in Problem 30, assume that there is no damping (i.e., b=0), m=1 and k=1. To see the effect of aging consider as positive parameter.

(a) Redo Problem 30with b=0and 畏arbitrary but fixed.

(b) Set 畏 =0 in the expansion obtained in part (a). Does this expansion agree with the expansion for the solution to the problem with 畏=0. [Hint: When 畏 =0 the solution is x(t)=cos t].
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