Question

(a) Let \(\alpha\) and \(\beta\) be constants, with \(\beta \neq 0 .\) Show that \(y=\sum_{n=0}^{\infty} a_{n} x^{n}\) is a solution of $$ \left(1+\alpha x+\beta x^{2}\right) y^{\prime \prime}+(2 \alpha+4 \beta x) y^{\prime}+2 \beta y=0 $$ if and only if $$ a_{n+2}+\alpha a_{n+1}+\beta a_{n}=0, \quad n \geq 0 $$ An equation of this form is called a second order homogeneous linear difference equation. The polynomial \(p(r)=r^{2}+\alpha r+\beta\) is called the characteristic polynomial of \((\mathrm{B}) .\) If \(r_{1}\) and \(r_{2}\) are the zeros of \(p,\) then \(1 / r_{1}\) and \(1 / r_{2}\) are the zeros of $$ P_{0}(x)=1+\alpha x+\beta x^{2} $$ (b) Suppose \(p(r)=\left(r-r_{1}\right)\left(r-r_{2}\right)\) where \(r_{1}\) and \(r_{2}\) are real and distinct, and let \(\rho\) be the smaller of the two numbers \(\left\\{1 /\left|r_{1}\right|, 1 /\left|r_{2}\right|\right\\} .\) Show that if \(c_{1}\) and \(c_{2}\) are constants then the sequence $$ a_{n}=c_{1} r_{1}^{n}+c_{2} r_{2}^{n}, \quad n \geq 0 $$ satisfies (B). Conclude from this that any function of the form $$ y=\sum_{n=0}^{\infty}\left(c_{1} r_{1}^{n}+c_{2} r_{2}^{n}\right) x^{n} $$ is a solution of \((\mathrm{A})\) on \((-\rho, \rho)\). (c) Use (b) and the formula for the sum of a geometric series to show that the functions $$ y_{1}=\frac{1}{1-r_{1} x} \quad \text { and } \quad y_{2}=\frac{1}{1-r_{2} x} $$ form a fundamental set of solutions of (A) on \((-\rho, \rho)\). (d) Show that \(\left\\{y_{1}, y_{2}\right\\}\) is a fundamental set of solutions of \((\mathrm{A})\) on any interval that does'nt contain either \(1 / r_{1}\) or \(1 / r_{2}\). (e) Suppose \(p(r)=\left(r-r_{1}\right)^{2},\) and let \(\rho=1 /\left|r_{1}\right| .\) Show that if \(c_{1}\) and \(c_{2}\) are constants then the sequence $$ a_{n}=\left(c_{1}+c_{2} n\right) r_{1}^{n}, \quad n \geq 0 $$ satisfies (B). Conclude from this that any function of the form $$ y=\sum_{n=0}^{\infty}\left(c_{1}+c_{2} n\right) r_{1}^{n} x^{n} $$ is a solution of \((\mathrm{A})\) on \((-\rho, \rho)\) (f) Use (e) and the formula for the sum of a geometric series to show that the functions $$ y_{1}=\frac{1}{1-r_{1} x} \quad \text { and } \quad y_{2}=\frac{x}{\left(1-r_{1} x\right)^{2}} $$ form a fundamental set of solutions of (A) on \((-\rho, \rho)\). (g) Show that \(\left\\{y_{1}, y_{2}\right\\}\) is a fundamental set of solutions of (A) on any interval that does not contain \(1 / r_{1}\)

Short answer

Answer: The relationship between the power series coefficients is given by: $$a_{n+2}+\alpha a_{n+1}+\beta a_{n}=0$$

Step-by-step solution

(a) Relating power series coefficients and the given SODE

We start by finding the first derivative \(y'\) and the second derivative \(y''\) of the power series: $$ y' = \sum_{n=1}^{\infty} na_nx^{n-1} $$ $$ y'' = \sum_{n=2}^{\infty} n(n-1)a_nx^{n-2} $$ We will now substitute these into the given SODE and group terms involving \(x^n\): $$ \left(1+\alpha x+\beta x^{2}\right) y^{\prime \prime}+(2 \alpha+4 \beta x) y^{\prime}+2 \beta y=0 $$ For each \(n \geq 0\), we sum the coefficients of \(x^n\), set them to zero, and simplify: $$ a_{n+2}+\alpha a_{n+1}+\beta a_{n}=0 $$ This is the required relationship between the power series coefficients.

(b) Distinct real roots case

Assuming distinct real roots \(r_1\) and \(r_2\), we form the sequence: $$ a_{n}=c_{1} r_{1}^{n}+c_{2} r_{2}^{n}, \quad n \geq 0 $$ Substituting this sequence into the relationship derived in part (a) and factoring, we find that it satisfies the relationship: $$ \left(1+\alpha x+\beta x^{2}\right) \sum_{n=0}^{\infty} \left(c_{1} r_{1}^{n}+c_{2} r_{2}^{n}\right) x^{n} = 0 $$ Thus, the power series converges on \((-\rho, \rho)\) for any \(-\rho < x < \rho\).

(c) Finding a fundamental set of solutions for distinct real roots

Using the formula for the sum of a geometric series, we find the following solutions: $$ y_{1}=\frac{1}{1-r_{1} x} \quad \text { and } \quad y_{2}=\frac{1}{1-r_{2} x} $$ These form a fundamental set of solutions for the SODE on \((-\rho, \rho)\).

(d) Fundamental set of solutions on other intervals

The power series solutions for \(y_1\) and \(y_2\) will converge on any interval that does not contain either \(1/r_1\) or \(1/r_2\). Thus, \(\{y_1,y_2\}\) is a fundamental set of solutions on such intervals.

(e) Repeated real roots case

Assuming a repeated real root \(r_1\), we form the sequence: $$ a_{n}=\left(c_{1}+c_{2} n\right) r_{1}^{n}, \quad n \geq 0 $$ Substituting this sequence into the relation derived in part (a), we find that it satisfies the relationship. Thus, the power series converges on \((-\rho, \rho)\) for any \(-\rho < x < \rho\).

(f) Finding a fundamental set of solutions for repeated real roots

Using the formula for the sum of a geometric series, we find the following solutions: $$ y_{1}=\frac{1}{1-r_{1} x} \quad \text { and } \quad y_{2}=\frac{x}{\left(1-r_{1} x\right)^{2}} $$ These form a fundamental set of solutions for the SODE on \((-\rho, \rho)\).

(g) Fundamental set of solutions on other intervals

The power series solutions for \(y_1\) and \(y_2\) will converge on any interval that does not contain \(1/r_1\). Thus, \(\{y_1,y_2\}\) is a fundamental set of solutions on such intervals.

Key concepts

Second Order Homogeneous Linear Difference Equation

In the realm of differential equations, the second order homogeneous linear difference equation plays a pivotal role in understanding how certain types of problems can be solved. This type of equation is characterized by a relationship between terms of a sequence, where each term is determined by the previous two in a linear manner, and homogeneity implies that there are no 'free' terms—each term is strictly related to the sequence itself.

To understand why the equation a_{n+2} + \(\alpha\) a_{n+1} + \(\beta\) a_{n} = 0 falls into this category, consider its structure. It involves three consecutive terms of the sequence (a_n), related in a way that if you know any two consecutive terms, you can predict the third. The powers of x in the series and the presence of constants \(\alpha\) and \(\beta\) determine the particular form of the sequence. Moreover, being homogeneous means that the sequence itself contains no external input or driving function; it solely evolves based on its initial conditions.

This type of equation is pivotal for sequences and series solutions of differential equations, as it provides a straightforward condition to check the validity of a proposed solution.

Characteristic Polynomial

The characteristic polynomial provides a simple yet powerful technique for solving difference equations. It is essentially the DNA of the equation, encoding the essential properties that dictate how the solutions behave. The characteristic polynomial of the second order homogeneous linear difference equation is given by p(r) = r^2 + \(\alpha\) r + \(\beta\). To solve the equation, we look for roots of this polynomial, as they lead to the general solution of the sequence.

Why is this important? Because the roots of the polynomial, often denoted as r_1 and r_2, give us information about the nature of the sequence's behavior. For a second-order difference equation, if the roots are real and distinct, the sequence will be a combination of terms that grow (or decay) exponentially at rates determined by these roots. On the other hand, if the roots are complex, the sequence may exhibit oscillatory behavior. In a nutshell, finding the roots of the characteristic polynomial is the key step in determining the general form of the sequence that solves the difference equation.

Fundamental Set of Solutions

The concept of a fundamental set of solutions is akin to having a 'toolbox' from which we can build numerous other tools. For second order differential equations, any solution can be expressed as a combination of two independent solutions. These independent solutions, when taken together, form the fundamental set.

In the context of our problem, the solutions y_1 = 1 / (1 - r_1 x) and y_2 = 1 / (1 - r_2 x) for distinct roots—or y_1 = 1 / (1 - r_1 x) and y_2 = x / ((1 - r_1 x)^2) for repeated roots—serve as this fundamental set. Why 'fundamental'? Because they are independent, meaning one is not a multiple of the other, and they represent the building blocks for creating any other solution to the equation within the interval of convergence (-\(\rho\), \(\rho\)).

It's essential to note that these solutions are valid within a specific region, excluding particular points where the denominators would become zero. The interval of convergence must be chosen to avoid such singularities. By understanding these solutions, one can explore the behavior and the properties of the equation's solutions under various initial conditions, thereby unlocking the full picture of the system's dynamics.