Question

Derive the indicial equation for the Riemann DE.

Short answer

The indicial equation for Riemann's Differential Equation is obtained by substitution of a power series solution and equating coefficients to zero, resulting in \( r(r-1)c_0 + (ar+b)rc_0 + cc_0 = 0 \) for the smallest power of \(x\) (when \(n=0\))

Step-by-step solution

Write down the Riemann Differential Equation and assumed power series solution

The Riemann Differential Equation is given by \( x^2y'' + (ax+b)y' + cy = 0 \). Then, let's assume a solution of the form \( y = ∑_{n=0}^{∞} c_nx^{n+r} \).

Substitute power series solution into Riemann Differential Equation

Through substitution, the Riemann's equation transforms into \( x^2 ( ∑_{n=0}^{∞} (n+r)(n+r-1)c_nx^{n+r-2} ) + (ax+b) ( ∑_{n=0}^{∞} (n+r)c_nx^{n+r-1} ) + c ( ∑_{n=0}^{∞} c_nx^{n+r} ) = 0 \)

Equating Coefficients and Formulating the Indicial Equation

As all coefficients of the power series must be zero for the equality to hold, we equate coefficients to zero and formulate the indicial equation. For the smallest power of \(x\) (when \(n=0\)), the indicial equation will be given by \( r(r-1)c_0 + (ar+b)rc_0 + cc_0 = 0 \)

Key concepts

Power Series Solution

To solve the Riemann Differential Equation, we start by assuming a solution in the form of a power series. This series is expressed as:

• \( y = \sum_{n=0}^{\infty} c_n x^{n+r} \).

In this expression, \(c_n\) are the coefficients that we need to determine, and \(r\) is a constant introduced into the exponent. The power series is a powerful tool as it can help transform complex differential equations into simpler algebraic forms.

By substituting this series form into the original equation, we take advantage of the power of series expansions. Each term in the equation is expressed in terms of \(x\), allowing us to handle even cumbersome terms involving derivatives with ease. This approach reveals deeper insights about solutions, especially where classic methods might fail.

Overall, a power series solution opens the door for systematic methods of finding solutions to seemingly intractable differential equations. This strategy aligns with the goal of understanding how solutions can be represented as infinite sums, helping in finding exact or approximate solutions.

Indicial Equation

The indicial equation is a crucial part of the analysis when solving differential equations with power series. Upon substituting the assumed power series into the Riemann Differential Equation, the next step involves assessing the coefficients of each term.

These coefficients, derived from expanding the equation into a series form, need to be zero for the equality to hold true across different values of \(x\). The indicial equation is formed specifically from the terms with the smallest exponents of \(x\). In this context, for the smallest power, we equate the coefficient resulting in:

• \( r(r-1)c_0 + (ar+b)rc_0 + cc_0 = 0 \).

The indicial equation is vital for gaining insights about possible values for \(r\). Solving this quadratic equation gives key values, known as roots, which influence the indices of the series solution itself. In simpler terms, it helps us determine the behavior and properties of the solution near singular points of the differential equation.

Differential Equations

Differential equations, like the Riemann Differential Equation, describe relationships involving rates of changes and functions. These equations are foundational in mathematics, modeling real-world phenomena where continuous change occurs, such as physics, biology, and engineering.

The Riemann Differential Equation takes a particular structural form, yet is representative of many such second-order linear equations. This equation is characterized by its quintessential makeup that includes terms with functions, their first derivatives, and even second derivatives:

• \( x^2y'' + (ax+b)y' + cy = 0 \)

Here, \(a\), \(b\), and \(c\) are constants that influence the behavior and type of solutions. In handling these, techniques like power series solutions become indispensable.

Overall, understanding differential equations and their transformative methods, like power series, unleashes vast potential in investigating dynamic systems. It allows for a standardized approach to determining solutions to complex problems by utilizing mathematical structures and techniques.