Content check — Differential Equations (Lebl), Chapter 8: Power-series methods

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Chapter 8: Power-series methods

13 problems · 0 figures

Exercise 7.1.101 §7.1 source ↗

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

Answer

Yes. Radius of convergence is $10$ ⁠.

Exercise 7.1.102 §7.1 source ↗

(challenging) Is $\sum_{n = 1}^{\infty} \frac{n!}{n^{n}} x^{n}$ a convergent power series? If so, find the radius of convergence.

Answer

Yes. Radius of convergence is $e$ ⁠.

Exercise 7.1.103 §7.1 source ↗

Using the geometric series, expand $\frac{1}{1 - x}$ around $x_{0} = 2$ ⁠. For what $x$ does the series converge?

Answer

$\frac{1}{1 - x} = - \frac{1}{1 - (2 - x)}$ so $\frac{1}{1 - x} = \sum_{n = 0}^{\infty} {(- 1)}^{n + 1} {(x - 2)}^{n}$ ⁠, which converges for $1 < x < 3$ ⁠.

Exercise 7.1.104 §7.1 source ↗

(challenging) Find the Taylor series for $x^{7} e^{x}$ around $x_{0} = 0$ ⁠.

Answer

$\sum_{n = 7}^{\infty} \frac{1}{(n - 7)!} x^{n}$

Exercise 7.1.105 §7.1 source ↗

(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 $f (x) - g (x)$ ⁠?

Answer

$f (x) - g (x)$ is a polynomial. Hint: Use Taylor series.

Exercise 7.1.106 §7.1 source ↗

Reindex the following series to have the powers be $x^{k}$ ⁠.

a) $\sum_{k = 3}^{\infty} k (k - 1) x^{k + 3}$

b) $\sum_{k = 1}^{\infty} (k + 1) x^{k - 1}$

c) $\sum_{k = 5}^{\infty} 2 k x^{k - 2}$

Answer

a) $\sum_{k = 6}^{\infty} (k - 3) (k - 4) x^{k}$ b) $\sum_{k = 0}^{\infty} (k + 2) x^{k}$ c) $\sum_{k = 3}^{\infty} 2 (k + 2) x^{k}$

Exercise 7.2.101 §7.2 source ↗

Use power series methods to solve $y^{''} + 2 x^{3} y = 0$ at the point $x_{0} = 0$ ⁠.

Answer

$a_{2} = 0$ ⁠, $a_{3} = 0$ ⁠, $a_{4} = 0$ ⁠, recurrence relation (for $k \geq 5$ ⁠): $a_{k} = \frac{- 2 a_{k - 5}}{k (k - 1)}$ ⁠, so:
$y (x) =$ $a_{0} +$ $a_{1} x -$ $\frac{a_{0}}{10} x^{5} -$ $\frac{a_{1}}{15} x^{6} +$ $\frac{a_{0}}{450} x^{10} +$ $\frac{a_{1}}{825} x^{11} -$ $\frac{a_{0}}{47250} x^{15} -$ $\frac{a_{1}}{99000} x^{16} +$ $\dots$

Exercise 7.2.102 §7.2 source ↗

(challenging) Power-series methods also work for 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 $y (0) = 0$ ⁠, $y^{'} (0) = 0$ ⁠.

Answer

a) $a_{2} = \frac{1}{2}$ ⁠, and for $k \geq 3$ we have $a_{k} = \frac{a_{k - 3} + 1}{k (k - 1)}$ ⁠, so
$y (x) =$ $a_{0} +$ $a_{1} x +$ $\frac{1}{2} x^{2} +$ $\frac{a_{0} + 1}{6} x^{3} +$ $\frac{a_{1} + 1}{12} x^{4} +$ $\frac{3}{40} x^{5} +$ $\frac{a_{0} + 7}{180} x^{6} +$ $\frac{a_{1} + 13}{504} x^{7} +$ $\frac{43}{2240} x^{8} +$ $\frac{a_{0} + 187}{12960} x^{9} +$ $\frac{a_{1} + 517}{45360} x^{10} +$ $\dots$
b) $y (x) =$ $\frac{1}{2} x^{2} +$ $\frac{1}{6} x^{3} +$ $\frac{1}{12} x^{4} +$ $\frac{3}{40} x^{5} +$ $\frac{7}{180} x^{6} +$ $\frac{13}{504} x^{7} +$ $\frac{43}{2240} x^{8} +$ $\frac{187}{12960} x^{9} +$ $\frac{517}{45360} x^{10} +$ $\dots$

Exercise 7.2.103 §7.2 source ↗

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?

Answer

Applying the method of this section directly we obtain $a_{k} = 0$ for all $k$ and so $y (x) = 0$ is the only solution we find.

Exercise 7.3.101 §7.3 source ↗

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

a) $y^{''} + y = 0$

b) $x^{3} y^{''} + (1 + x) y = 0$

c) $x y^{''} + x^{5} y^{'} + y = 0$

d) $\sin (x) y^{''} - y = 0$

e) $\cos (x) y^{''} - \sin (x) y = 0$

Answer

a) ordinary, b) singular but not regular singular, c) regular singular, d) regular singular, e) ordinary.

Exercise 7.3.102 §7.3 source ↗

Find the general solution of $x^{2} y^{''} - y = 0$ ⁠.

Answer

$y = A x^{\frac{1 + \sqrt{5}}{2}} + B x^{\frac{1 - \sqrt{5}}{2}}$

Exercise 7.3.103 §7.3 source ↗

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

Answer

$y =$ $x^{3 / 2}$ $\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{k! (k + 2)!} x^{k}$ (Note that for convenience we did not pick $a_{0} = 1$ ⁠.)

Exercise 7.3.104 §7.3 source ↗

(tricky) Find the general solution of $x^{2} y^{''} - x y^{'} + y = 0$ ⁠.

Answer

$y = A x + B x \ln (x)$

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Exercises and answers are adapted from Notes on Diffy Qs: Differential Equations for Engineers by Jiří Lebl, which is free to read and download, and are used under the CC BY-NC-SA 4.0 licence. They are taken from the book’s LaTeX source and set for the web, so they may differ slightly from the printed page. This adaptation is released under the same licence and is neither affiliated with nor endorsed by Lebl or Jiří Lebl.