Question

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. $$3x{y}^{\prime \prime }+(3x+1)y^{\prime }+y=0$$

Short answer

Find the indicial equation and solve for r. Substitute the series into the differential equation and determine recurrence relation. Solve for coefficients a_n.

Step-by-step solution

Understand the Frobenius Method

The Frobenius method is a technique to find power series solutions of linear differential equations near a singular point. First, find the recurrence relation for the series coefficients.

Rewrite the Differential Equation in Standard Form

Express the differential equation in the form: $$y^{″}+p(x)y^{\prime }+q(x)y=0$$ so that the coefficients are functions of x. For the given equation: $$3x{y}^{″}+(3x+1)y^{\prime }+y=0$$ divide by 3x to obtain $$y^{″}+\left(\frac{3x+1}{3x}\right)y^{\prime }+\frac{1}{3x}y=0$$

Assume a Frobenius Series Solution

Assume a solution of the form: $$y=\sum _{n=0}^{f}ty{a}_n{x}^{n+r}$$ where r is a number to be determined. Derive expressions for y', and y''.

Find Expressions for Derivatives

Take the derivatives of the assumed series: $$y^{\prime }=\sum _{n=0}^{f}ty{a}_n(n+r)x^{n+r-1}$$ and $$y^{″}=\sum _{n=0}^{f}ty{a}_n(n+r)(n+r-1)x^{n+r-2}$$

Substitute Series into the Differential Equation

Substitute the series expressions into the rewritten differential equation: $$\sum _{n=0}^{f}ty{a}_n(n+r)(n+r-1)x^{n+r-1}+\frac{3x+1}{3x}\sum _{n=0}^{f}ty{a}_n(n+r)x^{n+r-2}+\frac{1}{3x}\sum _{n=0}^{f}ty{a}_n{x}^{n+r}=0$$

Simplify and Collect Terms

Multiply to remove fractions, combine like terms, and set the coefficient of each power of x to 0. Solve for the recursion relation among the coefficients a_n.

Solve the Indicial Equation

The indicial equation is obtained from the lowest power term of x. Solve this equation to find the possible values of r. These values of r are the exponents in the Frobenius method.

Find the General Recurrence Relation

Use the recurrence relation found to express the coefficients a_n as functions of a_0 and a_1. This will give the power series solution in terms of a general parameter.

Write the General Solution

Combine all the terms to express the general solution to the differential equation as the sum of the two linearly independent solutions obtained.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. They play a vital role in modeling many real-world application problems in physics, engineering, biology, and economics. In our exercise, we are dealing with a linear second-order differential equation:

$$3x{y}^{″}+(3x+1)y^{\prime }+y=0$$

This equation is challenging to solve directly and requires specific methods, like the Frobenius method, to find the solution. Differential equations can be ordinary (ODE), involving a single variable, or partial (PDE), involving multiple variables. The one provided here is an ODE, specifically a homogeneous linear second-order ODE.

Power Series Solution

A power series solution is a technique to solve differential equations by assuming the solution can be expressed as an infinite series. The Frobenius method assumes a solution of the form:

$$y=\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^{n+r}$$

where $a_n$ are coefficients and $r$ is a constant. The step-by-step approach helps us systematically find these coefficients. By taking derivatives and substituting back into the original differential equation, we can simplify and find relationships among the coefficients.

This method is particularly useful when solving differential equations around singular points. A singular point is where the solution may become undefined or discontinuous. By expanding the solution in a series, we can express the behavior of the differential equation around these points.

Recurrence Relation

A recurrence relation is an equation that recursively defines the terms of a sequence. In the Frobenius method, after substituting the power series into the differential equation and equating coefficients of like powers of $x$, we derive a recurrence relation for the coefficients $a_n$.

This relationship allows us to express $a_{n+1}$ in terms of previous coefficients $a_n$, forming a sequence. For example, we might find that:

$$a_{n+1}=f(a_n)$$

The recurrence relation is crucial because it determines each coefficient in the series from the initial conditions. Ultimately, it helps us define the entire power series solution comprehensively.

Indicial Equation

The indicial equation is derived from the coefficient of the lowest power of $x$ after substituting the series back into the differential equation. It helps determine the possible values of $r$ in our assumed series solution.

Solving the indicial equation gives the exponent positions, which are critical for the Frobenius method. For our given differential equation:

$$3x{y}^{″}+(3x+1)y^{\prime }+y=0$$

Upon simplifying, we might find an equation like:

$$r(r-1)+\text{terms}=0$$

Solving this quadratic equation provides values for $r$, which are then used to find the series solution. In many cases, we get two values for $r$, indicating two independent solutions. Combining these solutions, we get the general solution for the differential equation.