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Chapter 1: Q- 26E (page 1)

Question: In Problems 23鈥�26, express the given power series as a series with generic term.

26.鈭�n=1鈭�ann+3xn+3

Short Answer

Expert verified

The required term is 鈭�k=4鈭�ak-3kxk

Step by step solution

01

Power series.

A power series is an infinite series of the form,

鈭�n=0鈭�an(x-c)n=a0+a1(x-c)+a2(x-c)2+.....

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

02

To express the given series in terms of the generic term

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)=鈭�n=1鈭�ann+3xn+3.

Let,

n+3=kn=k-3

So,

role="math" localid="1664278859581" 鈭�n=1鈭�ann+3xn+3=鈭�k-3=1鈭�ak-3kxk=鈭�k=4鈭�ak-3kxk

Hence, the required term is鈭�k=4鈭�ak-3kx.

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Most popular questions from this chapter

In Project C of Chapter 4, it was shown that the simple pendulum equation胃''(t)+蝉颈苍胃(t)=0 has periodic solutions when the initial displacement and velocity show that the period of the solution may depend on the initial conditions by using the vectorized Runge鈥揔utta algorithm with h= 0.02 to approximate the solutions to the simple pendulum problem on

[0, 4] for the initial conditions:

localid="1664100454791" (a)鈥�鈥�胃(0)=0.1,胃'(0)=0(b)鈥�鈥�胃(0)=0.5,胃'(0)=0(c)鈥�鈥�胃(0)=1.0,胃'(0)=0

[Hint: Approximate the length of time it takes to reach].

(a) For the initial value problem (12) of Example 9. Show that 蠒1(x)=0 and蠒2(x)=(x-2)3are solutions. Hence, this initial value problem has multiple solutions. (See also Project G in Chapter 2.)

(b) Does the initial value problemy'=3y23,y(0)=10-7have a unique solution in a neighbourhood ofx=0?

Newton鈥檚 law of cooling states that the rate of change in the temperature T(t) of a body is proportional to the difference between the temperature of the medium M(t) and the temperature of the body. That is, dTdt=KMt-Ttwhere K is a constant. Let K=0.04min-1and the temperature of the medium be constant, Mt=293kelvins. If the body is initially at 360 kelvins, use Euler鈥檚 method with h = 3.0 min to approximate the temperature of the body after

(a) 30 minutes.

(b) 60 minutes.

In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

y-logey=x2+1,dydx=2xyy-1

The directional field for dydx=4xyin shown in figure 1.12.

(a) Verify that the straight lines y=卤2xare solution curves, provided x鈮�0.

(b) Sketch the solution curve with initial condition y (0) = 2.

(c) Sketch the solution curve with initial condition y(2) = 1.

(d) What can you say about the behaviour of the above solution as x鈫�+鈭�? How about x鈫�-鈭�?
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