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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: Q9E (page 252)

Find two power series solutions of the given differential equation about the ordinary point.

Short Answer

Expert verified

Answer:

The solution is;

Step by step solution

01

Given information

The given equation is.

02

Identify all the power series.

03

Substitute our power series for the original equation.

04

Find first, second and third power series

First power series

Second power series

Third power series

05

Combine the three power series

Locate the first term of the first power series withas the index. We integrate the three power series and extract a common factorafter discovering the term.

06

Find a recurrence relation.

07

Find and .

,

08

conclusion

The final solution is:

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Most popular questions from this chapter

In Problems 23 and 24 use the procedure in Example 8 to find two power series solutions of the given differential equation about the ordinary point X = 0.

$y'' + (s i n x) y = 0$

In Problems 15–24, x = 0is a regular singular point of the givendifferential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on $0, \infty$ . $2 xy " - y' + 2 y = 0$

In problem 9 and 10 use (18) to find the general solution of the given differential equation on

$(0, \infty)$

Question 10:

$x^{2} y'' + xy' - (2 x^{2} + 64) = 0$

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.

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.

$e^{- x / 2}$

See all solutions

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