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Chapter 1: Q-10E (page 1)

Question 10: In Problems, find the power series expansion for f(x)+g(x), given the expansions for f(x) and g(x).

10.

Short Answer

Expert verified

The required expansion is,

Step by step solution

01

Power series expansion 

We know that the power series representation of the sum of two power series, given byis simply obtained by term-wise addition:

For the given series

02

 changing of index

In order to obtain sum we need to have common index for both the series f(x) and g(x).

Changing the index of f(x). Let z = n - 1 = > n = z + 1

Since index is just a dummy variable, we can replace z with n; the series f(x) thus becomes

Changing the index of g(x),

If we substitute other variable in g(x), it will change the order of the variable (x-1), therefore,

The summation of the series f(x)+g(x) is therefore represented by

Hence, the required expansion is,

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Most popular questions from this chapter

Let c >0. Show that the function ϕ(x)=(c2-x2)-1is a solution to the initial value problemdydx=2xy2,y(0)=1c2,on the interval-c<x<c. Note that this solution becomes unbounded as x approaches ±c. Thus, the solution exists on the interval(-δ,δ) with δ=c, but not for largerδ. This illustrates that in Theorem 1, the existence interval can be quite small (IFC is small) or quite large (if c is large). Notice also that there is no clue from the equationdydx=2xy2 itself, or from the initial value, that the solution will “blow up” atx=±c.

Question: Let f(x)and g(x)be analytic at x0. Determine whether the following statements are always true or sometimes false:

(a) 3f(x)+g(x)is analytic at x0.

(b) f(x)/g(x)is analytic at x0.

(c) f'(x)is analytic at x0.

(d) f(x)3-∫x0xg(t)dt is analytic at x0.

(a) For the initial value problem (12) of Example 9. Show that ϕ1(x)=0 andϕ2(x)=(x-2)3are solutions. Hence, this initial value problem has multiple solutions. (See also Project G in Chapter 2.)

(b) Does the initial value problemy'=3y23,y(0)=10-7have a unique solution in a neighbourhood ofx=0?

In problems 1-6, identify the independent variable, dependent variable, and determine whether the equation is linear or nonlinear.

t3d3xdt32+3t-x=0

In Problems 21–26, solve the initial value problem.

(y2sinx)dx+(1x-yx)dy,y(π)=1
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