Answered: determine whether x = 0 is an ordinary…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Concept explainers

Question

determine whether x = 0 is an ordinary point or a regular singular point of the given differential equation. Then obtain two linearly independent solutions to the differential equation and state the maximum interval on which your solutions are valid.

Q. xy′′+2y′+xy=0

Expert Solution

Step 1

Consider the differential equation:

$x y'' + 2 y' + x y = 0$

Divide by x throughout.

$y'' + \frac{2}{x} y' + y = 0$

Compare with,

$y'' + P (x) y' + Q (x) y = 0$

So,

$P (x) = \frac{2}{x} Q (x) = 1$

Since, $P (x) \to \infty$ at $x = 0$ , therefore $x = 0$ is a singular point.

Now,

$\begin{array}{rcl} x P (x) & = & 2 \\ x^{2} Q (x) & = & x^{2} \end{array}$

Since both are finite at $x = 0$ , therefore $x = 0$ is a regular singular point.

Step 2

Suppose, the power series solution is:

$y = \sum_{n = 0}^{\infty} c_{n} x^{n + r}$

Differentiate with respect to x.

$\begin{array}{rcl} y' & = & \sum_{n = 0}^{\infty} (n + r) c_{n} x^{n + r - 1} \\ y'' & = & \sum_{n = 0}^{\infty} (n + r) (n + r - 1) c_{n} x^{n + r - 2} \end{array}$

Substitute into the differential equation.

$\begin{array}{rcl} x \sum_{n = 0}^{\infty} (n + r) (n + r - 1) c_{n} x^{n + r - 2} + 2 \sum_{n = 0}^{\infty} (n + r) c_{n} x^{n + r - 1} + x \sum_{n = 0}^{\infty} c_{n} x^{n + r} & = & 0 \\ \sum_{n = 0}^{\infty} (n + r) (n + r - 1) c_{n} x^{n + r - 1} + \sum_{n = 0}^{\infty} 2 (n + r) c_{n} x^{n + r - 1} + \sum_{n = 0}^{\infty} c_{n} x^{n + r + 1} & = & 0 \\ \sum_{n = 0}^{\infty} (n + r) (n + r - 1) c_{n} x^{n + r - 1} + \sum_{n = 0}^{\infty} 2 (n + r) c_{n} x^{n + r - 1} + \sum_{n = 2}^{\infty} c_{n - 2} x^{n + r - 1} & = & 0 \\ [r (r - 1) c_{0} + 2 r c_{0}] x^{r - 1} + [r (r + 1) c_{1} + 2 (r + 1) c_{1}] x^{r} + \sum_{n = 2}^{\infty} [(n + r) (n + r - 1) c_{n} + 2 (n + r) c_{n} + c_{n - 2}] x^{n + r - 1} & = & 0 \end{array}$

Step by step

Solved in 4 steps

Knowledge Booster

$Differential Equation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated