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Chapter 12: Q18 P (page 605)

To study the approximations in the table, a computer plot on the same axes the given function together with its small approximation and its asymptotic approximation. Use an interval large enough to show the asymptotic approximation agreeing with the function for large . If the small approximation is not clear, plot it alone with the function over a small interval y2(x).

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01

The Concept of Wolfram Mathematica Built-in Functions

The Wolfram Mathematica built-in functions are used because they are very convenient. The Mathematica function Bessel J[p, x]returns the values of JP(x).

02

Use the Wolfram Mathematica built-in function to plot the graphs.

The approximate size for small is yn(x)=-(2n+1)!!xn+1+Ox1-n,.

The approximate formula for largeis,yn(x)=-1xcosx-nπ2+Ox-2.

The plot of the functions together with its small approximation and its asymptotic approximation can be drawn as:

The plot of the functions y2(x) together with its small x' approximation over a small interval can be drawn as:


The plot of the functions y2(x)together with its asymptotic approximation over a large interval can be drawn as:


It can be seen that the asymptotic approximation agrees with the function of large x .

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Most popular questions from this chapter

Solve the following eigenvalue problem (see end of Section 2 and problem 11): Given the differential equation y''+(λx−14−l(l+1)x2)y=0where l is an integerlocalid="1654860659044" ≥0 , find values of localid="1654860714122" λsuch that localid="1654860676211" y→0 aslocalid="1654860742759" role="math" x→∞ , and find the corresponding eigenfunctions. Hint: letlocalid="1654860764612" y=xl+1e−x/2v(x), and show that localid="1654860784518" v(x) satisfies the differential equationlocalid="1654860800910" xv''+(2l+2−x)v'+(λ−l−1)v=0.Comparelocalid="1654860829619" (22.26) to show that if localid="1654860854431" λ is an integerlocalid="1654860871428" >l, there is a polynomial solution localid="1654860888067" v(x)=Lλ−t−12t+1(x).Solve the eigenvalue problem localid="1654860910472" y''+(λx−14−l(l+1)x2)y=0.

For Problems 1 to 4, find one (simple) solution of each differential equation by series, and then find the second solution by the "reduction of order" method, Chapter 8, Section 7 (e).

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

To study the approximations in the table, computer plot on the same axes the given function together with its small x approximation and its asymptotic approximation. Use an interval large enough to show the asymptotic approximation agrees with the function for large x. If the small x approximation is not clear, plot it alone with the function over a small interval J2(x).


For Problems 1 to 4 , find one (simple) solution of each differential equation by series, and then find the second solution by the "reduction of order" method, Chapter 8 , Section 7 (e).

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

From (17.4), show that,hn(1)(ix)=-e-x/x .
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