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Chapter 12: Problem 2

Solve the following differential equations by the method of Frobenius (generalized power series). Remember that the point of doing these problems is to learn about the method (which we will use later), not just to find a solution. You may recognize some series [as we did in (11.6)] or you can check your series by expanding a computer answer. $$x^{2} y^{\prime \prime}+x y^{\prime}-9 y=0$$

Short Answer

Expert verified
The general solution is given by: \[ y = C_1 x^3 Σ_{n=0}^{∞}a_n x^n + C_2 x^{-3} Σ_{n=0}^{∞}a_n x^n \]where \(a_n\) are the coefficients obtained from the series.

Step by step solution

01

Verify the standard form

First, confirm that the given differential equation is in the standard form for applying the Frobenius method. The equation is already in the standard form: \[ x^2 y'' + x y' - 9 y = 0 \]
02

Find the indicial equation

Assume a solution of the form: \[ y = \textstyle\begin{displaystyle} \textstyle\begin{displaystyle} \textstyle\begin{displaystyle} \textstyle\begin{displaystyle} \textstyle\begin{displaystyle} x^r \textstyle\begin{displaystyle} \textstyle\begin{displaystyle}\textstyle\begin{displaystyle} \textstyle\begin{displaystyle} \textstyle\begin{displaystyle} \textstyle\begin{displaystyle} \textstyle\begin{displaystyle}\textstyle\begin{displaystyle} \textstyle\begin{displaystyle}Σ_{n=0}^{\textstyle\begin{displaystyle}∞}a_n x^n \] Compute the derivatives: \[ y' = \textstyle\begin{displaystyle} Σ_{n=0}^{\textstyle\begin{displaystyle}\textstyle\begin{displaystyle}∞}(n+r)a_n x^{n+r-1} \] \[ y'' = \textstyle\begin{displaystyle} Σ_{n=0}^{\textstyle\begin{displaystyle}∞}(n+r)(n+r-1)a_n x^{n+r-2} \]
03

Substitute the series into the differential equation

Substitute the series and its derivatives into the differential equation:\[ x^2 \textstyle\begin{displaystyle} Σ_{n=0}^{∞}(n+r)(n+r-1)a_n x^{n+r-2} + x \textstyle\begin{displaystyle} Σ_{n=0}^{∞}(n+r)a_n x^{n+r-1} - 9 \textstyle\begin{displaystyle} Σ_{n=0}^{∞}a_n x^{n+r} = 0 \] Combine like terms: \[ Σ_{n=0}^{∞}(n+r)(n+r-1)a_n x^{n+r} + Σ_{n=0}^{∞}(n+r)a_n x^{n+r} - 9 Σ_{n=0}^{∞}a_n x^{n+r} = 0 \]
04

Solve the indicial equation

Equate the coefficients of \( x^r \) to zero: \[ (n+r)(n+r-1)a_n + (n+r)a_n - 9a_n = 0 \] Simplify to form the indicial equation: \[ r(r-1) + r - 9 = 0 \] This simplifies to: \[ r^2 - 9 = 0 \] Solve for \( r \): \[ r = 3 \text{ or } r = -3 \]
05

Construct the series solution

For each value of \( r \), construct the series solution:- For \( r = 3 \): \[ y_1 = x^3 Σ_{n=0}^{∞}a_n x^n \] - For \( r = -3 \): \[ y_2 = x^{-3} Σ_{n=0}^{∞}a_n x^n \] These solutions represent the general form of the Frobenius series solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Differential Equations
Differential equations are equations that involve an unknown function and its derivatives. They are used to model various physical, biological, and economic phenomena. An important subset of these equations is linear second-order differential equations. For example, consider the differential equation provided:

\[ x^2 y'' + x y' - 9 y = 0 \]
In this equation, y is the unknown function of x, and y' and y'' are the first and second derivatives of y, respectively. The objective is to find the function y(x) that satisfies this equation. Various methods exist to solve such equations, and the Frobenius method, in particular, is useful when dealing with singular points.
The Role of the Indicial Equation
The indicial equation plays a crucial role when using the Frobenius method to solve differential equations. It helps in determining the exponents, denoted as r, at the singular point. Here’s how you typically start:

Assume a solution of the form:
\[ y = x^r \sum_{n=0}^{\infty} a_n x^n \]

When you plug this assumed solution into the differential equation, you end up with terms that depend on r. The process leads to a polynomial equation in terms of r, which is what we call the indicial equation. In this specific exercise:

Statements like \( x^2 y'' + x y' - 9y =0 \) lead us to form the indicial equation:
\[ r(r-1) + r - 9 = 0 \]

Solving the indicial equation gives:
\[ r = 3 \text{ or } r = -3 \]

These roots, r = 3 and r = -3, are fundamental to constructing the full series solution.
Constructing the Series Solution
The series solution is built upon the roots found from the indicial equation. For each root, you obtain a different solution that contributes to the general solution of the differential equation. Let’s break it down:

For r = 3, assume:
\[ y_1 = x^3 \sum_{n=0}^{\infty} a_n x^n \]

And for r = -3, assume:
\[ y_2 = x^{-3} \sum_{n=0}^{\infty} a_n x^n \]

These assumptions help in converting the original differential equation into a simpler form involving series terms. Summing up these series solutions gives the general form:

\[ y(x) = c_1 x^3 \sum_{n=0}^{\infty} a_n x^n + c_2 x^{-3} \sum_{n=0}^{\infty} a_n x^n \]

The constants c1 and c2 are determined by the initial or boundary conditions of the problem. By substituting the series back into the differential equation, you can ensure that it satisfies the original equation, providing verification for the derived solutions.

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Most popular questions from this chapter

Solve the following differential equations by series and also by an elementary method and verify that your solutions agree. Note that the goal of these problems is not to get the answer (that's easy by computer or by hand) but to become familiar with the method of series solutions which we will be using later. Check your results by computer. $$x y^{\prime}=x y+y$$

Show that the functions \(e^{i n \pi x / l}, n=0,\pm 1,\pm 2, \cdots,\) are a set of orthogonal functions on \((-l, l)\).

Solve the following eigenvalue problem (see end of Section 2 and Problem 11): Given the differential equation $$y^{\prime \prime}+\left(\frac{\lambda}{x}-\frac{1}{4}-\frac{l(l+1)}{x^{2}}\right) y=0$$ where \(l\) is an integer \(\geq 0,\) find values of \(\lambda\) such that \(y \rightarrow 0\) as \(x \rightarrow \infty\), and find the corresponding eigenfunctions. Hint: let \(y=x^{l+1} e^{-x / 2} v(x),\) and show that \(v(x)\) satisfies the differential equation $$x v^{\prime \prime}+(2 l+2-x) v^{\prime}+(\lambda-l-1) v=0$$ Compare (22.26) to show that if \(\lambda\) is an integer \(>l,\) there is a polynomial solution \(v(x)=L_{\lambda-l-1}^{2 l+1}(x)\).

Consider each of the following problems as illustrations showing that, in a power series solution, we must be cautious about using the general recursion relation between the coefficients for the first few terms of the series. Solve \(y^{\prime \prime}+y^{\prime} / x^{2}=0\) by power series to find the relation $$a_{n+1}=-\frac{n(n-1)}{n+1} a_{n}.$$ If, without thinking carefully, we test the series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) for convergence by the ratio test, we find $$\lim _{n \rightarrow \infty} \frac{\left|a_{n+1} x^{n+1}\right|}{\left|a_{n} x^{n}\right|}=\infty\quad (Show this.)$$ Thus we might conclude that the series diverges and that there is no power series solution of this equation. Show why this is wrong, and that the power series solution is \(y=\) const.

Solve the following differential equations by the method of Frobenius (generalized power series). Remember that the point of doing these problems is to learn about the method (which we will use later), not just to find a solution. You may recognize some series [as we did in (11.6)] or you can check your series by expanding a computer answer. $$2 x y^{\prime \prime}+y^{\prime}+2 y=0$$
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