7.E: Power series methods (Exercises)

7.1: Power Series

Q7.1.1

Is the power series \( \sum_{k=0}^\infty e^k x^k\) convergent? If so, what is the radius of convergence?

Q7.1.2

Is the power series \( \sum_{k=0}^\infty k x^k\) convergent? If so, what is the radius of convergence?

Q7.1.3

Is the power series \( \sum_{k=0}^\infty k! x^k\) convergent? If so, what is the radius of convergence?

Q7.1.4

Is the power series \( \sum_{k=0}^\infty \frac{1}{(2k)!} {(x-10)}^k\) convergent? If so, what is the radius of convergence?

Q7.1.5

Determine the Taylor series for \(\sin x\) around the point \(x_0 = \pi\).

Q7.1.6

Determine the Taylor series for \(\ln x\) around the point \(x_0 = 1\), and find the radius of convergence.

Q7.1.7

Determine the Taylor series and its radius of convergence of \(\dfrac{1}{1+x}\) around \(x_0 = 0\).

Q7.1.8

Determine the Taylor series and its radius of convergence of \(\dfrac{x}{4-x^2}\) around \(x_0 = 0\). Hint: You will not be able to use the ratio test.

Q7.1.9

Expand \(x^5+5x+1\) as a power series around \(x_0 = 5\).

Q7.1.10

Suppose that the ratio test applies to a series \( \sum_{k=0}^\infty a_k x^k\). Show, using the ratio test, that the radius of convergence of the differentiated series is the same as that of the original series.

Q7.1.11

Suppose that \(f\) is an analytic function such that \(f^{(n)}(0) = n\). Find \(f(1)\).

Q7.1.101

Is the power series \( \sum_{n=1}^\infty {(0.1)}^n x^n\) convergent? If so, what is the radius of convergence?

Q7.1.102

[challenging] Is the power series \( \sum_{n=1}^\infty \frac{n!}{n^n} x^n\) convergent? If so, what is the radius of convergence?

Q7.1.103

Using the geometric series, expand \(\frac{1}{1-x}\) around \(x_0=2\). For what \(x\) does the series converge?

Q7.1.104

[challenging] Find the Taylor series for \(x^7 e^x\) around \(x_0 = 0\).

Q7.1.105

[challenging] Imagine \(f\) and \(g\) are analytic functions such that \(f^{(k)}(0) = g^{(k)}(0)\) for all large enough \(k\). What can you say about \(f(x)-g(x)\)?

7.2: Series solutions of linear second order ODEs

In the following exercises, when asked to solve an equation using power series methods, you should find the first few terms of the series, and if possible find a general formula for the \(k^{\text{th}}\) coefficient.

Q7.2.1

Use power series methods to solve \(y''+y = 0\) at the point \(x_0 = 1\).

Q7.2.2

Use power series methods to solve \(y''+4xy = 0\) at the point \(x_0 = 0\).

Q7.2.3

Use power series methods to solve \(y''-xy = 0\) at the point \(x_0 = 1\).

Q7.2.4

Use power series methods to solve \(y''+x^2y = 0\) at the point \(x_0 = 0\).

Q7.2.5

The methods work for other orders than second order. Try the methods of this section to solve the first order system \(y'-xy = 0\) at the point \(x_0 = 0\).

Q7.2.6

(Chebyshev’s equation of order \(p\)): a) Solve \((1-x^2)y''-xy' + p^2y = 0\) using power series methods at \(x_0=0\). b) For what \(p\) is there a polynomial solution?

Q7.2.7

Find a polynomial solution to \((x^2+1) y''-2xy'+2y = 0\) using power series methods.

Q7.2.8

a) Use power series methods to solve \((1-x)y''+y = 0\) at the point \(x_0 = 0\). b) Use the solution to part a) to find a solution for \(xy''+y=0\) around the point \(x_0=1\).

Q7.2.101

Use power series methods to solve \(y'' + 2 x^3 y = 0\) at the point \(x_0 =0\).

Q7.2.102

[challenging] We can also use power series methods in nonhomogeneous equations. a) Use power series methods to solve \(y'' - x y = \frac{1}{1-x}\) at the point \(x_0 = 0\). Hint: Recall the geometric series. b) Now solve for the initial condition \(y(0)=0\), \(y'(0) = 0\).

Q7.2.103

Attempt to solve \(x^2 y'' - y = 0\) at \(x_0 = 0\) using the power series method of this section (\(x_0\) is a singular point).

Can you find at least one solution? Can you find more than one solution?

7.3: Singular points and the method of Frobenius

Q7.3.3

Find a particular (Frobenius-type) solution of \(x^2 y'' + x y' + (1+x) y = 0\).

Q7.3.4

Find a particular (Frobenius-type) solution of \(x y'' - y = 0\).

Q7.3.5

Find a particular (Frobenius-type) solution of \(y'' +\frac{1}{x}y' - xy = 0\).

Q7.3.6

Find the general solution of \(2 x y'' + y' - x^2 y = 0\).

Q7.3.7

Find the general solution of \(x^2 y'' - x y' -y = 0\).

Q7.3.8

In the following equations classify the point \(x=0\) as ordinary, regular singular, or singular but not regular singular.

1. \(x^2(1+x^2)y''+xy=0\)
2. \(x^2y''+y'+y=0\)
3. \(xy''+x^3y'+y=0\)
4. \(xy''+xy'-e^xy=0\)
5. \(x^2y''+x^2y'+x^2y=0\)

Q7.3.101

In the following equations classify the point \(x=0\) as ordinary, regular singular, or singular but not regular singular.

1. \(y''+y=0\)
2. \(x^3y''+(1+x)y=0\)
3. \(xy''+x^5y'+y=0\)
4. \(\sin(x)y''-y=0\)
5. \(\cos(x)y''-\sin(x)y=0\)

Q7.3.102

Find the general solution of \(x^2 y'' -y = 0\).

Q7.3.103

Find a particular solution of \(x^2 y'' +(x-\frac{3}{4})y = 0\).

Q7.3.3

[tricky] Find the general solution of \(x^2 y'' - x y' +y = 0\).