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A First Course In Differential Equations With Modelling Applications

Dennis G. Zill

$Math Studyset Vaia Explanations$ Math

11th Edition · 400 pages

ISBN: 9781305965720

Chapter 6: Q39E (page 243)

In Problem 19, find an easier way than multiplying two power series to obtain the Maclaurin series representation of

Short Answer

Expert verified

Answer:

The Maclaurin series ofis:

Step by step solution

01

Given Information

The function is

02

To obtain the Maclaurin series

We substitute (1) and (2) into the given differential equation.

Using double angle formula, to solve this problem

By substituting (2) into (1), we get

Guided by the information in Table 2, the Maclaurin series of the sin function is

By replacingwithwe get

03

Substitute the equation

Therefore, the Maclaurin series of theis

Hence, the Maclaurin series ofwhich is:

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Most popular questions from this chapter

In Problems 11-16 use an appropriate series in (2) to find the Maclaurin series of the given function. Write your answer in summation notation.

$e^{- x / 2}$

In Problems 21 and 22 the given function is analytic at $a = 0$ . Use appropriate series in (2) and long division to find the first four nonzero terms of the Maclaurin series of the given function.

$s e c x$

Cooling Fin A cooling fin is an outward projection from a mechanical or electronic device from which heat can be radiated away from the device into the surrounding medium (such as air). See Figure 6.R.1. An annular, or ring-shaped, cooling fin is normally used on cylindrical surfaces such as a circular heating pipe. See Figure 6.R.2. In the latter case, let r denote the radial distance measured from the center line of the pipe and T(r) the temperature within the fin defined for $r_{0} \leq r \leq r_{1}$ It can be shown that T(r) satisfies the differential equation

role="math" localid="1663927167728" $\frac{d}{d r} (r \frac{d T}{d r}) = a^{2} r (T - T_{m})$

where a²is a constant and T_mis the constant air temperature.

Suppose $r_{-} \{0\} = 1, r_{-} \{1\} = 3,$ ,and T_m=70. Use the substitution w(r) =T(r)^_70to show that the solution of the given differential equation subject to the boundary conditions

T(1)=160, T(3)=0 is

role="math" localid="1663926265607" $T (r) = 70 + 90 \frac{K_{1} (3 a) I_{v} (a r) + I_{1} (3 a) K_{v} (a r)}{K_{1} (3 a) I_{0} (a) + I_{!} (3 a) K_{0} (a)}$ where and I₀(x) and K₀(x)are the modified Bessel functions of the first and second kind. You will also have to use the derivatives given in (25) of Section 6.4.

Find two power series solutions of the given differential equation about the ordinary point $x = 0$ as $y'' + x^{2} y' + x y = 0 .$

In Problems, 7-18 find two power series solutions of the given differential equation about the ordinary point $(x^{2} - 1) y^{''} + x y^{'} - y = 0$

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