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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset Vaia Explanations$ Math

9th Edition · 616 pages

ISBN: 9780321977069

Chapter 1: Q-12E (page 1)

Question: In Problem, find the first three nonzero terms in the power series expansion for the product f(x) g(x).

Short Answer

Expert verified

The required product is,

Step by step solution

Given power series expansion

The series expansion of the given series is given by

Find the product of the power series expansion

Multiplying both the series

The first three non-zero terms of the product are,

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Most popular questions from this chapter

Question: The Taylor series for f(x) =ln (x)about x₂=0given in equation (13) can also be obtained as follows:

(a)Starting with the expansion 1/ (1-s) = $\sum_{n = 0}^{\infty} s''$ and observing that

obtain the Taylor series for 1/xabout x₀= 1.

(b)Since use the result of part (a) and termwise integration to obtain the Taylor series for f (x)=lnxaboutx₀= 1.

Pendulum with Varying Length. A pendulum is formed by a mass m attached to the end of a wire that is attached to the ceiling. Assume that the length l(t)of the wire varies with time in some predetermined fashion. If

U(t) is the angle in radians between the pendulum and the vertical, then the motion of the pendulum is governed for small angles by the initial value problem $l^{2} (t) θ'' (t) + 2 l (t) l' (t) θ' (t) + gl (t) \sin (θ (t)) = 0; θ (0) = θ_{o}, θ' (0) = θ_{1}$ where g is the acceleration due to gravity. Assume that $l (t) = l_{o} + l_{1} \cos (ωt - ϕ)$ where $l_{1}$ is much smaller than $l_{o}$ . (This might be a model for a person on a swing, where the pumping action changes the distance from the center of mass of the swing to the point where the swing is attached.) To simplify the computations, take g = 1. Using the Runge– Kutta algorithm with h = 0.1, study the motion of the pendulum when $θ_{o} = 0.05, θ_{1} = 0, l_{o} = 1, l_{1} = 0.1, ω = 1, ϕ = 0.02$ . In particular, does the pendulum ever attain an angle greater in absolute value than the initial angle $θ_{o}$ ?

Verify that the function $ϕ (x) = c_{1} e^{x} + c_{2} e^{- 2 x}$ is a solution to the linear equation $\frac{d^{2} y}{{dx}^{2}} + \frac{dy}{dx} - 2 y = 0$ for any choice of the constants $c_{1}$ and $c_{2}$ . Determine $c_{1}$ and $c_{2}$ so that each of the following initial conditions is satisfied.

(a) $y (0) = 2, y' (0) = 1$

(b) $y (1) = 1, y' (1) = 0$

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.

(d) ${[f (x)]}^{3} - \int_{x_{0}}^{x} g (t) d t$ is analytic at x₀.

Using the Runge–Kutta algorithm for systems with h = 0.05, approximate the solution to the initial value problem $y''' + y'' + y^{2} = t; y (0) = 1, y' (0) = 0, y'' (0) = 1$ at t=1.

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Question: In Problem, find the first three nonzero terms in the power series expansion for the product f(x) g(x).

Short Answer

Step by step solution

Given power series expansion

Find the product of the power series expansion

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Most popular questions from this chapter

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