Try the fastest way to create flashcards
Question

# For the differential equations, find the indicial polynomial for the singularity at x=0. Then find the recurrence formula for the largest of the roots to the indicial equation. xy''+(1-x)y'-y=0

Solution

Verified
Step 1
1 of 4

$\textbf{Solution}$: Let us consider the given differential equation

\begin{align}xy''+(1-x)y'-y=0.\end{align}

Now notice that, if we recasted the above equation we have

\begin{align*} xy''+(1-x)y'-y=0 \implies \> &x^2y''+(1-x)xy'-xy=0.\end{align*}

Let us now consider the general Euler's equation as

$x^2y''+pxy'+qy=0,\>\>\>\> \text{ where p and q are constants}.$

Comparing above two equations we have

$p=1-x\>\>\> \text{ and }\>\>\> q=-x.$

From the aforesaid argument we have, both $p$ and $q$ are analytic at $x=0$. So, $x=0$ is a regular singular point. Let us now look at the Frobenius solution as

$y(x)=\sum_{n=0}^{\infty} a_n x^{s+n}.$

Now notice that

\begin{align*} y'(x) &= \sum_{n=0}^{\infty} (s+n) a_n x^{s+n-1} \\ y''(x) &= \sum_{n=0}^{\infty} (s+n)(s+n-1) a_n x^{s+n-2}. \end{align*}

## Recommended textbook solutions

#### Differential Equations with Boundary Value Problems

2nd EditionISBN: 9780131862364 (1 more)Al Boggess, David Arnold, John Polking
2,356 solutions

#### Fundamentals of Differential Equations

9th EditionISBN: 9780321977069 (5 more)Arthur David Snider, Edward B. Saff, R. Kent Nagle
2,119 solutions

#### A First Course in Differential Equations with Modeling Applications

10th EditionISBN: 9781111827052 (2 more)Dennis G. Zill
2,369 solutions

#### Differential Equations with Boundary-Value Problems

9th EditionISBN: 9781305965799 (3 more)Dennis G. Zill
3,184 solutions