Articles

9: Series Solutions of ODEs (Frobenius’ Method)


  • 9.1: Frobenius’ Method
    The Frobenius method is a method to identify an infinite series solution for a second-order ordinary differential equation.
  • 9.2: Singular Points
    Typically, the Frobenius method identifies two independent solutions provided that the indicial equation's roots are not separated by an integer.
  • 9.3: Special Cases
    For the two special cases I will just give the solution. It requires a substantial amount of algebra to study these two cases.

9: Series Solutions of ODEs (Frobenius’ Method)

Let us look at the a very simple (ordinary) di󻀎rential equation,

with initial conditions y ( 0 ) = a , y &prime ( 0 ) = b . Let us assume that there is a solution that is analytical near t = 0 . This means that near t = 0 the function has a Taylor&rsquos series

y ( t ) = c 0 + c 1 t + &hellip = &sum k = 0 &infin c k t k . (9.2)

(We shall ignore questions of convergence.) Let us proceed

y &prime ( t ) = c 1 + 2 c 2 t + &hellip = &sum k = 1 &infin k c k t k &minus 1 , y &Prime ( t ) = 2 c 2 + 3 &sdot 2 t + &hellip = &sum k = 2 &infin k ( k &minus 1 ) c k t k &minus 2 , t y ( t ) = c 0 t + c 1 t 2 + &hellip = &sum k = 0 &infin c k t k + 1 . (9.3)

Combining this together we have

y &Prime &minus t y = [ 2 c 2 + 3 &sdot 2 t + &hellip ] &minus [ c 0 t + c 1 t 2 + &hellip ] = 2 c 2 + ( 3 &sdot 2 c 3 &minus c 0 ) t + &hellip = 2 c 2 + &sum k = 3 &infin k ( k &minus 1 ) c k &minus c k &minus 3 t k &minus 2 . (9.4)

Here we have collected terms of equal power of t . The reason is simple. We are requiring a power series to equal 0 . The only way that can work is if each power of x in the power series has zero coe󻀼ient. (Compare a finite polynomial. ) We thus find

c 2 = 0 , k ( k &minus 1 ) c k = c k &minus 3 . (9.5)

The last relation is called a recurrence of recursion relation, which we can use to bootstrap from a given value, in this case c 0 and c 1 . Once we know these two numbers, we can determine c 3 , c 4 and c 5 :

c 3 = 1 6 c 0 , c 4 = 1 1 2 c 1 , c 5 = 1 2 0 c 2 = 0 . (9.6)

These in turn can be used to determine c 6 , c 7 , c 8 , etc. It is not too hard to find an explicit expression for the c &rsquos

c 3 m = 3 m &minus 2 ( 3 m ) ( 3 m &minus 1 ) ( 3 m &minus 2 ) c 3 ( m &minus 1 ) = 3 m &minus 2 ( 3 m ) ( 3 m &minus 1 ) ( 3 m &minus 2 ) 3 m &minus 5 ( 3 m &minus 3 ) ( 3 m &minus 4 ) ( 3 m &minus 5 ) c 3 ( m &minus 1 ) = ( 3 m &minus 2 ) ( 3 m &minus 5 ) &hellip 1 ( 3 m ) ! c 0 , c 3 m + 1 = 3 m &minus 1 ( 3 m + 1 ) ( 3 m ) ( 3 m &minus 1 ) c 3 ( m &minus 1 ) + 1 = 3 m &minus 1 ( 3 m + 1 ) ( 3 m ) ( 3 m &minus 1 ) 3 m &minus 4 ( 3 m &minus 2 ) ( 3 m &minus 3 ) ( 3 m &minus 4 ) c 3 ( m &minus 2 ) + 1 = ( 3 m &minus 2 ) ( 3 m &minus 5 ) &hellip 2 ( 3 m + 1 ) ! c 1 , c 3 m + 1 = 0 . (9.7)

The general solution is thus

y ( t ) = a 1 + &sum m = 1 &infin c 3 m t 3 m + b 1 + &sum m = 1 &infin c 3 m + 1 t 3 m + 1 . (9.8)

The technique sketched here can be proven to work for any di󻀎rential equation

y &Prime ( t ) + p ( t ) y &prime ( t ) + q ( t ) y ( t ) = f ( t ) (9.9)

provided that p ( t ) , q ( t ) and f ( t ) are analytic at t = 0 . Thus if p , q and f have a power series expansion, so has y .

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9: Series Solutions of ODEs (Frobenius’ Method)

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Mathematical Expression Editor

We continue our study of the method of Frobenius for finding series solutions of linear second order differential equations, extending to the case where the indicial equation has a repeated real root.

The Method of Frobenius II

In this section we discuss a method for finding two linearly independent Frobenius solutions of a homogeneous linear second order equation near a regular singular point in the case where the indicial equation has a repeated real root. As in the preceding section, we consider equations that can be written as

where . We assume that the indicial equation has a repeated real root . In this case Theorem thmtype:7.5.3 implies that (eq:7.6.1) has one solution of the form but does not provide a second solution such that is a fundamental set of solutions. The following extension of Theorem thmtype:7.5.2 provides a way to find a second solution.

Proof Theorem thmtype:7.5.2 implies (eq:7.6.4). Differentiating formally with respect to in (eq:7.6.3) yields

To prove that satisfies (eq:7.6.6), we view in (eq:7.6.2) as a function of two variables, where the prime indicates partial differentiation with respect to thus, With this notation we can use (eq:7.6.2) to rewrite (eq:7.6.4) as

Proof Since is a repeated root of , the indicial polynomial can be factored as so which is nonzero if . Therefore the assumptions of Theorem thmtype:7.6.1 hold with , and (eq:7.6.4) implies that . Since it follows that , so (eq:7.6.6) implies that This proves that and are both solutions of . We leave the proof that is a fundamental set as an exercise.

and Computing recursively with (eq:7.6.13) and (eq:7.6.14) yields and Substituting these coefficients into (eq:7.6.10) yields and

Since the recurrence formula (eq:7.6.11) involves three terms, it’s not possible to obtain a simple explicit formula for the coefficients in the Frobenius solutions of (eq:7.6.9). However, as we saw in the preceding sections, the recurrence formula for involves only two terms if either or in (eq:7.6.1). In this case, it’s often possible to find explicit formulas for the coefficients. The next two examples illustrate this.

To obtain in (eq:7.6.17), we must compute for . We’ll do this by logarithmic differentiation. From (eq:7.6.18), Therefore Differentiating with respect to yields Therefore Setting here and recalling (eq:7.6.19) yields

Since (eq:7.6.20) can be rewritten as Therefore, from (eq:7.6.17),

To obtain in (eq:7.6.23), we must compute for . Again we use logarithmic differentiation. From (eq:7.6.24), Taking logarithms yields Differentiating with respect to yields Therefore Setting and recalling (eq:7.6.25) yields

Since (eq:7.6.26) can be rewritten as Substituting this into (eq:7.6.23) yields

If the solution of reduces to a finite sum, then there’s a difficulty in using logarithmic differentiation to obtain the coefficients in the second solution. The next example illustrates this difficulty and shows how to overcome it.

To obtain in (eq:7.6.29) we must compute for . Let’s first try logarithmic differentiation. From (eq:7.6.30), so Differentiating with respect to yields Therefore

However, we can’t simply set here if , since the bracketed expression in the sum corresponding to contains the term . In fact, since for , the formula (eq:7.6.31) for is actually an indeterminate form at .

We overcome this difficulty as follows. From (eq:7.6.30) with , Therefore so

From (eq:7.6.30) with , where Therefore which implies that if . We leave it to you to verify that Substituting this and (eq:7.6.32) into (eq:7.6.29) yields

Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)


9: Series Solutions of ODEs (Frobenius’ Method)

In this chapter we will finally be looking at nonconstant coefficient differential equations. While we won’t cover all possibilities in this chapter we will be looking at two of the more common methods for dealing with this kind of differential equation.

The first method that we’ll be taking a look at, series solutions, will actually find a series representation for the solution instead of the solution itself. You first saw something like this when you looked at Taylor series in your Calculus class. As we will see however, these won’t work for every differential equation.

The second method that we’ll look at will only work for a special class of differential equations. This special case will cover some of the cases in which series solutions can’t be used.

Here is a brief listing of the topics in this chapter.

Review : Power Series – In this section we give a brief review of some of the basics of power series. Included are discussions of using the Ratio Test to determine if a power series will converge, adding/subtracting power series, differentiating power series and index shifts for power series.

Review : Taylor Series – In this section we give a quick reminder on how to construct the Taylor series for a function. Included are derivations for the Taylor series of (<f e>^) and (cos(x)) about (x = 0) as well as showing how to write down the Taylor series for a polynomial.

Series Solutions – In this section we define ordinary and singular points for a differential equation. We also show who to construct a series solution for a differential equation about an ordinary point. The method illustrated in this section is useful in solving, or at least getting an approximation of the solution, differential equations with coefficients that are not constant.

Euler Equations – In this section we will discuss how to solve Euler’s differential equation, (ax^<2>y'' + b x y' +c y = 0). Note that while this does not involve a series solution it is included in the series solution chapter because it illustrates how to get a solution to at least one type of differential equation at a singular point.


THE GEOMETRY OF MATHEMATICAL METHODS

In this section, we will briefly discuss the theorem that states when a second order linear ode has power series solutions.

First, write the ode in the form:

and look at the functions (p(z)) and (q(z)) has function of the complex variable (z ext<.>) If (p(z)) and (q(z)) are analytic at a point (z=z_0 ext<,>) the (z_0) is said to be a regular point of the differential equation. (The word analytic is a technical term for a complex-valued function which is (complex) differentiable at the point. You can learn more about this concept online. But for practical purposes, it means that the function does not blow up at the point (z_0 ext<,>) nor is it otherwise badly behaved, e.g. the origin of a square root.)

Theorem 9.10.1 .

If the coefficients (p(z)) and (q(z)) are analytic at a point (z_0 ext<,>) then a power series solution of the differential equation (9.10.1), expanded around the point (z_0) exists, and furthermore, the radius of convergence for the series extends at least as far as the nearest singularity (point of non-analyticity) of (p(z)) or (q(z)) in the complex plane. Usually, there will be two such power series solutions, but sometimes the second solution will be a power series times a logarithm.

Definition 9.10.2 .

If ((z-z_0) p(z)) and ((z-z_0)^2 q(z)) are analytic, then the point (z_0) is called a regular singular point or regular singularity. Theorem: If (z_0) is a regular singular point then equation (9.10.1) can be solved by an extension of power series methods called a Frobenius Series. The solution will consist of: (1) two Frobenius series, or (2) one Frobenius series (y_1(z-z_0)) and a second solution (y_2(z-z_0)=y_1(z-z_0)ln(z-z_0)+y_0(z-z_0) ext<,>) where (y_0(z-z_0)) is a second Frobenius series. We will not discuss this method further here, but again, you can look it up online or in a more comprehensive mathematical methods text, if necessary.


Finding ODE series solution coefficients

I am trying to solve an ODE by subbing in a series form and then looking individually at the coefficients of different powers of the variable. I'm looking at a general form of equation:

I want to fill in the solution form $w(z) = e^ z^mu sum_^infty a_s z^<-s>$ and then collect coefficients of the various powers of $z$ . However when I try to do this the series terms are not being multiplied with any $z$ terms which are multiplying the series from the outside. A simpler example of the problem I'm having is when I input say:

The output of this is to give the coefficient as: $e^ z^5sum_^infty a_s z^<-s>$

What I want is for the $z^5$ to multiply the $z^mu$ term and also term-by-term with the series terms, and then give me just the coefficient of the resulting $z^mu$ term. So the only corresponding term in the sum is the one of the form $e^z^5z^mu a_5 z^<-5>$ , and therefore the coefficient I should be getting is just: $e^ a_<5>$

Can I not do this type of manipulation or have I just done something wrong here?


ME681A

Mathematics For Engineers

Credits:

Course Content:

Linear Algebra: Vector space and its basis Matrices as coordinate-dependent linear transformation null and range spaces Solution of linear algebraic equations: Gauss elimination and Gauss-Jordon methods, LU Decomposition and Cholesky method, Gauss-Seidel/ Jacobi iterative methods Condition number Minimum norm and least square error solutions Eigenvalues and eigenvectors of matrices and their properties Similarity transformation Jordon canonical form and orthogonal diagonalization Mises power method for finding eigenvalues/eigenvectors of symmetric matrices. Tensor Algebra and Index Notation. Vector and Tensor Calculus: Curves and surfaces Gradient, divergence and curl, Line, surface and volume integrals Gauss (divergence), Stokes and Green’s theorems. Topics in Numerical Methods: Solution of a non-linear algebraic equation and system of equations Interpolation methods, Regression Numerical Integration. Ordinary Differential Equations (ODEs): Techniques of the separation of variable and the integrating factor for 1st order ODEs Solutions of linear, 2nd order ODEs with constant coefficients and Euler-Cauchy ODEs System of 1st order ODEs Numerical methods for solving ODEs, Homogeneous, linear, 2nd order ODEs with variable coefficients: power series and Frobenius methods Sturm-Louville problem Laplace transform method for non-homogeneous, linear, 2nd order ODEs: discontinuous right-hand sides

Lecturewise Breakup (based on 50min per lecture)

I. Introduction (1 Lecture)

Introduction to the course. [1 Lecture]

II. Linear Algebra (12 Lectures)

Vector spaces: definition, linear independence of vectors, basis, inner product and inner product space, orthogonality, Gram-Schmidt procedure, subspaces. [2 Lectures]

Matrices: coordinate-dependent linear transformations, null and range spaces. [1 Lecture]

Linear algebraic equations: existence and uniqueness of solution, elementary row/column operations, Gauss elimination and Gauss Jordon methods, Echelon form, pivoting, LU decomposition and Cholesky method, Gauss-Seidel and Jacobi iterative methods, condition number, minimum norm and least square error solutions. [4.5 Lectures]

Eigenvalues and eigenvectors of matrices: properties like multiplicity, eigenspace, spectrum and linear independence of eigenvectors, similarity transformation and Jordon canonical form, eigenvalues/eigenvectors of symmetric matrices: orthogonal diagonalization. [3 Lectures]

Iterative methods to find eigenvalues/eigenvectors of symmetric matrices: forward iteration and Mises power method, inverse iteration. [1.5 Lectures]

III. Tensor Algebra (4 Lectures)

Index Notation and Summation Convention. [1 Lecture]

Tensor algebra: tensor as a linear vector transformation, dyadic representation, transformation of components, product of tensors, transpose, decomposition into symmetric and antisymmetric parts, invariants, decomposition into isotropic and deviatoric parts, inner product and norm, inverse, orthogonal tensors, eigenvalues and eigenvectors, square-root, positive definite symmetric tensor, polar decomposition, tensors of higher order. [3 Lectures]

IV. Vector and Tensor Calculus (4 Lectures)

Review of multi-variable calculus. [0.5 Lecture]

Curves: parametric representation, tangent vector, arc length, curvature, principal normal vector, osculating plane, bi-normal vector Surfaces: parametric representation, tangent vector and tangent plane. [1.5 Lectures]

Scalar fields: gradient, directional derivative, potential Vector fields: divergence, curl, solenoidal and irrotational vector fields Line integral and path independence Surface and volume integrals Gauss (divergence), Stokes and Green’s theorems (without proof). [1.5 Lectures]

Tensor calculus: tensor derivative of a scalar field, gradient of a vector field, divergence of a tensor field. [0.5 Lecture]

V. Topics in Numerical Methods (5 Lectures)

Solution of non-linear algebraic equations, Newton-Raphson method for a system of non-linear algebraic equations. [1.5 Lectures]

Lagrange and Hermite interpolation methods. [1 Lecture]

Regression: linear least-square method. [1 Lecture]

Numerical integration: trapezoidal and Simpson’s rules, Gauss quadrature. [1.5 Lectures]

VI. Ordinary Differential Equations (ODEs) (14 Lectures)

Initial value problem (IVP) of a 1st order ODE: existence, uniqueness and continuity with initial conditions. [0.5 Lecture]

Methods of solving 1st order ODEs: separation of variable technique, change of variable to make ODE separable exact ODEs, integrating factor to make ODE exact, linear 1st order ODEs. [1.5 Lectures]

Homogeneous, linear, 2nd order ODEs: existence and uniqueness, 2 fundamental (linearly independent) solutions and Wronskian, superposition for obtaining general solution, fundamental solutions of ODEs with constant coefficients, method of reduction of order to find 2nd linearly independent solution, fundamental solutions of Euler-Cauchy ODEs. [2 Lectures]

Non-homogeneous, linear, 2nd order ODEs: existence and uniqueness, methods of undermined coefficients and variation of parameters, introduction to higher order ODEs. [1.5 Lectures]

System of 1st order ODEs: existence and uniqueness of IVP, solution of the homogeneous system with constant coefficients, generalized eigenvector to find other fundamental solutions, method of variation of parameters.[1.5 Lectures]

Numerical methods for solving IVP of ODEs: Euler and Runge-Kutta methods, stability of numerical methods. [1.5 Lectures]

Homogeneous, linear, 2nd order ODEs with variable coefficients: power series method, solution of Legendre equation Frobenius method, solution of Bessel equation Sturm-Louville problem with regular, periodic and singular (homogeneous) boundary conditions and use of its eigenfunctions as an orthogonal basis for the representation of functions. [3.5 Lectures]

Laplace transform method for IV problem involving non-homogeneous, linear, 2nd order ODEs, properties of transform, inverse transform using tables, discontinuous right-hand sides involving unit step, impulse and Dirac-delta functions, t-shifting theorem. [2 Lectures]

References:

Advanced Engineering Mathematics by E. Kreyszig, John Wiley and Sons, International 8th Revised Edition, 1999,


MATH 263 Webwork Session

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MATHS -I - Mathematics-I (3110014) | GTU Syllabus (Old & Revised)

Teaching Hours

Indeterminate Forms and L'Haspital's Rule
Indeterminate Forms and L'Haspital's Rule.

Improper Integrals. Convergence and divergence of the integrals. Beta and Gamma finctions and their properties.

Applications of definite integral. Volume using cross-sections. Length of plane curves, Areas of Surfaces of Revolution

Convergence and divergence of sequences
Convergence and divergence of sequences, The Sandwich Theorem for Sequences. The Continuous Function Theorem for Sequences. Bounded Monotonic Sequences, Convergence and divergence of an infinite series, geometric series, telescoping series, n th term test for divergent series. Combining series, Harmonic Series, Integral test, The p - series, The Comparison test. The Limit Comparison test. Ratio test. Raabe's Test. Root test, Alternating series test, Absolute and Conditional convergence, Power series. Radius of convergence of a power series. Taylor and Maclaurin series.

Fourier Series
Fourier Series of 2PI periodic functions. Dirichlet's conditions for representation by a Fourier series. Orthogonality of the trigonometric system. Fourier Series of a function of period 2L.. Fourier Series of even and odd functions, Half range expansions.

Functions of several variables
Functions of several variables, Limits and continuity, Test for non existence of a limit. Partial differentiation. Mixed derivative theorem. differentiability, Chain rule, Implicit differentiation, Gradient, Directional derivative, tangent plane and normal line, total differentiation, Local extreme values, Method of Lagrange Multipliers.

Functions of several variables
Multiple integral, Double integral over Rectangles and general regions, double integrals as volumes. Change of order of integration, double integration in polar coordinates. Area by double integration, Triple integrals in rectangular, cylindrical and spherical coordinates. lacobian, multiple integral by substitution.

Elementary row operations in Matrix
Elementary row operations in Matrix, Row echelon and Reduced row echelon forms. Rank by echelon forms. Inverse by Gauss-Jordan method. Solution of system of linear equations by Gauss elimination and Gauss-Jordan methods. Eigen values and eigen vectors. Cayley-Hamilton theorem, Diagonal iration of a matrix.


9: Series Solutions of ODEs (Frobenius’ Method)

Ordinary Differential Equations

An Introduction to the Fundamentals

Prepared by Dr. Kenneth Howell,

Department of Mathematical Sciences, University of Alabama in Huntsville

Below are the chapters of the solution manual for Ordinary Differential Equations: An Introduction to the Fundamentals, published by CRC Press. (More precisely, below are the links to pdf files for the chapters.)

Some General Comments and Warnings:

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Q: Consider the 2nd order ODE: d y + ну %3D 0, хE (0, L] х€ [0, L] dx2 where, M, is a constant. The bou.

A: Idea: consider the three cases for values of u. then solve the differential equation.

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A: given that AD, BE and CF are altitudes H be orthocenter o.

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A: We know that a number is divisible by 3 iff sum of digit of number is by 3.

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Watch the video: ODE:: xy + y +2xy = 0:: Method of Frobenius Series Solution about a Regular Singular Point (November 2021).