7.4 Exercises

7.1: Power Series

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 $k^{th}$ coefficient.

Your Turn

(challenging)

We can also use power series methods in nonhomogeneous equations.

Use power series methods to solve $y^{″} - x y = \frac{1}{1 - x}$ at the point $x sub 0 equals 0$ . Hint: Recall the geometric series.
Now solve for the initial condition $y open parenthesis 0 close parenthesis equals 0$ , $y prime open parenthesis 0 close parenthesis equals 0$ .

Answer

$a_{2} = \frac{1}{2}$ , and for $k greater than or equal to 1$ we have $a_{k} = \frac{a_{k - 3} + 1}{k (k - 1)}$ , so $y (x) = a_{0} + a_{1} x + \frac{1}{2} x^{2} + \frac{a_{0} + 1}{6} x^{3} + \frac{a_{1} + 1}{12} x^{4} + \frac{3}{40} x^{5} + \frac{a_{0} + 2}{30} x^{6} + \frac{a_{1} + 2}{42} x^{7} + \frac{5}{112} x^{8} + \frac{a_{0} + 3}{72} x^{9} + \frac{a_{1} + 3}{90} x^{10} + \dots$
$y (x) = \frac{1}{2} x^{2} + \frac{1}{6} x^{3} + \frac{1}{12} x^{4} + \frac{3}{40} x^{5} + \frac{1}{15} x^{6} + \frac{1}{21} x^{7} + \frac{5}{112} x^{8} + \frac{1}{24} x^{9} + \frac{1}{30} x^{10} + \dots$

7.3: Singular points and the method of Frobenius

Your Turn

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

$x^{2} (1 + x^{2}) y^{″} + x y = 0$
$x^{2} y^{″} + y^{'} + y = 0$
$x y^{″} + x^{3} y^{'} + y = 0$
$x y^{″} + x y^{'} - e^{x} y = 0$
$x^{2} y^{″} + x^{2} y^{'} + x^{2} y = 0$

Your Turn

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

$y^{″} + y = 0$
$x^{3} y^{″} + (1 + x) y = 0$
$x y^{″} + x^{5} y^{'} + y = 0$
$\sin (x) y^{″} - y = 0$
$\cos (x) y^{″} - \sin (x) y = 0$

Answer

ordinary,
singular but not regular singular,
regular singular,
regular singular,
ordinary.

Your Turn

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

Answer

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

Adapted from Differential Equations for Engineers by Jiří Lebl (Oklahoma State University), hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.