Glossary

8.1 Power Series Solutions

3 min read•august 6, 2024

solutions are a powerful tool for solving differential equations when other methods fall short. They allow us to represent solutions as infinite sums, providing a way to analyze behavior near singular points.

In this section, we'll learn how to construct power series solutions, determine their convergence, and classify singular points. We'll also explore techniques like recurrence relations and the to solve differential equations using power series.

Power Series and Convergence

Fundamentals of Power Series

Top images from around the web for Fundamentals of Power Series

  • differential equations - Solving ODEs via power series - what is wrong with my solution ... View original

    Is this image relevant?

  • complex analysis - Convergence radius of power series for different centers - Mathematics Stack ... View original

    Is this image relevant?

  • differential equations - Solving ODEs via power series - what is wrong with my solution ... View original

    Is this image relevant?

1 of 3

Top images from around the web for Fundamentals of Power Series

  • differential equations - Solving ODEs via power series - what is wrong with my solution ... View original

    Is this image relevant?

  • complex analysis - Convergence radius of power series for different centers - Mathematics Stack ... View original

    Is this image relevant?

  • differential equations - Solving ODEs via power series - what is wrong with my solution ... View original

    Is this image relevant?

1 of 3
  • Power series express functions as an infinite sum of terms involving powers of a variable xx multiplied by coefficients ana_n
    • General form: ∑n=0∞an(x−c)n=a0+a1(x−c)+a2(x−c)2+⋯\sum_{n=0}^{\infty} a_n (x - c)^n = a_0 + a_1(x - c) + a_2(x - c)^2 + \cdots
    • cc represents the center of the power series
  • RR determines the interval on which a power series converges
    • Inside the interval (c−R,c+R)(c - R, c + R), the series converges
    • Outside this interval, the series diverges
    • On the boundary points x=c±Rx = c \pm R, further investigation is needed
  • Ordinary points of a differential equation are where the coefficients of the equation are analytic (can be represented by a convergent power series)

Classifying Singular Points

  • Regular singular points of a differential equation are where the coefficients have a pole of order at most 1
    • The point x=0x = 0 is a if the equation can be written as x2y′′+xp(x)y′+q(x)y=0x^2 y'' + xp(x)y' + q(x)y = 0, where p(x)p(x) and q(x)q(x) are analytic at x=0x = 0
    • Solutions near regular singular points can be represented by power series
  • Irregular singular points are where the coefficients have a pole of order greater than 1 or essential singularity
    • Power series solutions may not exist at irregular singular points

Solving Power Series

Recurrence Relation and Indicial Equation

  • Recurrence relations are equations that relate each coefficient ana_n to previous coefficients in a power series solution
    • Obtained by substituting the power series into the differential equation and equating coefficients of like powers of xx
    • Example: For the equation xy′′+y′−xy=0xy'' + y' - xy = 0, the is an+2=n−1(n+1)(n+2)ana_{n+2} = \frac{n - 1}{(n + 1)(n + 2)} a_n
  • The indicial equation determines the possible values of the power series exponent at a regular singular point
    • Found by substituting y=xry = x^r into the differential equation and solving for the lowest power of xx
    • Roots of the indicial equation give the values of rr, which determine the leading terms of the series solutions

Solving Techniques

  • The method of undetermined coefficients is used to solve linear differential equations with power series
    • Assume a solution of the form y=∑n=0∞anxny = \sum_{n=0}^{\infty} a_n x^n
    • Substitute the series into the differential equation and solve for the coefficients ana_n using the recurrence relation
  • solutions expand a function about a point x=cx = c using its derivatives
    • The coefficients are given by an=f(n)(c)n!a_n = \frac{f^{(n)}(c)}{n!}, where f(n)f^{(n)} denotes the nn-th derivative of ff
    • Example: The Taylor series for exe^x about x=0x = 0 is ex=∑n=0∞xnn!=1+x+x22!+x33!+⋯e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots

Key Terms to Review (16)

Analyticity: Analyticity refers to the property of a function that allows it to be represented by a power series in a neighborhood around a point. Functions that are analytic are infinitely differentiable within their radius of convergence, meaning they can be expressed as a Taylor series expansion that converges to the function in some interval. This concept is crucial because it connects differential equations to power series solutions, making it easier to analyze and solve them.
Augustin-Louis Cauchy: Augustin-Louis Cauchy was a French mathematician known for his foundational contributions to analysis and differential equations. His work laid the groundwork for many modern mathematical concepts, especially in relation to power series and their convergence, which is essential for finding solutions to differential equations.
Bernoulli Equation: The Bernoulli Equation refers to a specific type of differential equation that can be expressed in the form $$y' + P(x)y = Q(x)y^n$$, where $n$ is any real number other than 0 or 1. This equation is notable because it can be solved using a substitution that transforms it into a linear differential equation, making it easier to handle in various applications, particularly in fluid dynamics and other fields where growth processes are modeled.
Homogeneous Equations: Homogeneous equations are differential equations in which every term is a function of the dependent variable and its derivatives, typically having the form $a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + ... + a_1(x)y' + a_0(x)y = 0$. These equations exhibit certain properties that allow solutions to be expressed in terms of their linear combinations. A key aspect of homogeneous equations is that they can often be solved using power series solutions, which provide a systematic way to find solutions in the form of an infinite series.
Indicial Equation: An indicial equation is a polynomial equation that arises when solving differential equations with singular points, particularly in the context of power series and Frobenius methods. It is derived from substituting a power series solution into the differential equation, allowing for the determination of the leading term's exponent. This equation helps identify the possible values of this exponent, which are crucial for constructing valid solutions around singular points.
Initial Conditions: Initial conditions refer to the specific values assigned to the variables of a differential equation at a certain starting point, which are crucial for finding a unique solution. These conditions allow the mathematical model to describe real-world scenarios accurately by specifying the state of the system at the beginning of observation. They play a vital role in various applications, ensuring that solutions match physical phenomena or biological systems from a defined point in time.
Irregular singular point: An irregular singular point of a differential equation is a type of singular point where the behavior of solutions cannot be analyzed using standard power series methods. Unlike regular singular points, at irregular singular points, the solutions may not be expressible as a power series in a neighborhood of the point, making them more complex to study and understand. This intricacy arises because the coefficients of the differential equation do not have well-defined limits that allow for straightforward solution techniques.
Linear ordinary differential equations: Linear ordinary differential equations are equations that relate a function and its derivatives, and can be expressed in the form $$a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + ... + a_1(x)y' + a_0(x)y = g(x)$$ where the coefficients $$a_i(x)$$ are functions of the independent variable $$x$$ and $$g(x)$$ is a known function. This linearity means that the equation satisfies the superposition principle, making them easier to analyze and solve. They form a fundamental part of many mathematical models in physics, engineering, and other fields, often leading to solutions expressed as series or special functions.
Maclaurin Series: A Maclaurin series is a specific type of power series that represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point, specifically at zero. It is a special case of the Taylor series, which can be used to approximate functions and analyze their behavior near the origin. By using the Maclaurin series, one can express complicated functions in a simpler polynomial form, facilitating calculations and providing insights into the function's properties.
Ordinary differential equations: Ordinary differential equations (ODEs) are mathematical equations that involve functions of a single variable and their derivatives. These equations describe how a quantity changes over time or space and are essential in modeling real-world phenomena across various fields, including physics, engineering, biology, and economics. The solutions to ODEs provide critical insights into the behavior of dynamic systems, allowing us to predict future states based on initial conditions.
Power Series: A power series is an infinite series of the form $$ ext{S}(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + ...$$, where each term is a coefficient multiplied by a power of the variable x. Power series are significant in approximating functions, particularly when dealing with ordinary differential equations, as they allow for solutions that converge within a certain radius around a center point, typically denoted by the variable 'c'. They can represent a wide range of functions and can be manipulated through various operations such as differentiation and integration.
Radius of convergence: The radius of convergence is the distance from the center of a power series within which the series converges to a function. It’s crucial for understanding how power series can represent functions accurately, especially when solving differential equations using methods like power series solutions and the Frobenius method. This radius helps identify the limits within which the series can be relied upon for valid approximations and is influenced by the coefficients of the series.
Recurrence relation: A recurrence relation is a mathematical equation that defines a sequence of numbers using previous terms in the sequence. This concept is vital for finding solutions to various problems where current values depend on past values, making it a key tool in many mathematical fields. Recurrence relations are often used to derive specific solutions and can be related to different methods, such as power series or transformations, that help simplify complex problems into manageable forms.
Regular Singular Point: A regular singular point is a type of singularity in a differential equation where the coefficients of the equation behave in a controlled manner. At these points, the solution can still be expressed as a power series, and one can often apply the Frobenius method to find solutions. The existence of regular singular points allows us to classify the nature of the solutions and their behavior near these points, facilitating the use of series expansion techniques.
Taylor Series: A Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a single point. This mathematical representation allows for the approximation of complex functions as polynomials, making them easier to analyze and solve. The concept is crucial in various numerical methods and analytical techniques, as it provides a way to express functions locally around a point, thus linking theoretical mathematics with practical applications.
Uniform Convergence: Uniform convergence refers to a type of convergence of a sequence of functions where the speed of convergence is uniform across the entire domain. This means that for any given degree of accuracy, there exists a point in the sequence beyond which all function values are uniformly close to the limit function, regardless of where you look in the domain. This concept is important when discussing power series solutions, as it ensures that operations like differentiation and integration can be performed term-by-term without losing convergence.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide