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Chapter 1: Q-23E (page 1)

Question: In Problems 23–26, express the given power series as a series

with generic term Xk.

23.∑n=1∞nanxn-1

Short Answer

Expert verified

The required term is .∑k=0∞(k+1)ak+1xk.

Step by step solution

01

Power series

A power series is an infinite series of the form,

∑n=0∞an(x-c)n=a0+a1(x-c)+a2(x-c)2+....

Where, an represents the coefficient term of the nth term c, is a constant.

02

To express the given series in terms of generic term xk

In order to express the given series in terms of generic term xk. , we will change the index of the power series .

Given that,

f(x)=∑n=1∞nanxn-1

Let,

n-1=kn=k+1

So,

∑n=1∞nanxn-1=∑k+1=1∞(k+1)ak+1xk∑n=1∞nanxn-1=∑k=0∞(k+1)ak+1xk

Hence, the required term is∑k=0∞(k+1)ak+1xk

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Most popular questions from this chapter

In Problems 10–13, use the vectorized Euler method with h = 0.25 to find an approximation for the solution to the given initial value problem on the specified interval.

(1+t2)y''+y'-y=0;y(0)=1,y'(0)=-1on[0,1]

In Problems 21–26, solve the initial value problem (etx+1)dt+(et-1)dx=0,x(1)=1

Spring Pendulum.Let a mass be attached to one end of a spring with spring constant kand the other end attached to the ceiling. Letlo be the natural length of the spring, and let l(t) be its length at time t. Ifθ(t) is the angle between the pendulum and the vertical, then the motion of the spring pendulum is governed by the system

l''(t)-l(t)θ'(t)-gcosθ(t)+km(l-lo)=0l2(t)θ''(t)+2l(t)l'(t)θ'(t)+gl(t)sinθ(t)=0

Assume g = 1, k = m = 1, and lo= 4. When the system is at rest, l=lo+mgk=5.

a. Describe the motion of the pendulum when l(0)=5.5,l'(0)=0,θ(0)=0,θ'(0)=0.

b. When the pendulum is both stretched and given an angular displacement, the motion of the pendulum is more complicated. Using the Runge–Kutta algorithm for systems with h = 0.1 to approximate the solution, sketch the graphs of the length l and the angular displacement u on the interval [0,10] if l(0)=5.5,l'(0)=0,θ(0)=0.5,θ'(0)=0.

In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.

2x+yx-1dx+xy-1dy=0

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

dydx=3x-y-13,y(2)=1
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