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. $$x^{2} y^{\prime \prime}+2 x^{2} y^{\prime}-2 y=0$$

Short answer

The series solution for the differential equation is derived by substituting, solving the indicial equation, finding the recurrence relation, and determining coefficients.

Step-by-step solution

Identify the form of the differential equation

The given differential equation is

Identify the form of the differential equation

The given differential equation is

Form the indicial equation by substituting the series solution

We assume a solution of the form: Here, .

Find the recurrence relation

Substitute the series solutions where .

Solve the indicial equation

Solve for . This will give the values of the exponent .

Find the coefficients

Using the obtained recurrence relations and the values of , determine the coefficients .

Form the general solution

Write the general solution as a series expansion, which incorporates the obtained coefficients and exponents.

Verify the solution

Check the correctness by comparing the series to a known series expansion or using computational tools.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In simpler terms, they describe how a particular quantity changes over time or space. These equations are fundamental in fields like physics, engineering, and economics.

When solving differential equations, we often aim to find a function (or a set of functions) that satisfies the given equation. For example, in our problem, we have the differential equation:

\(x^{2} y^{\top \top}+2 x^{2} y^{\top}-2 y=0 \).

The solution to this equation represents a function \( y \) that depends on \( x \) and satisfies the equation for any value of \( x \). The Frobenius method is one powerful technique used to find series solutions to these types of equations.

Series Solution

The series solution approach involves expressing the solution to a differential equation as an infinite sum of terms. This is particularly useful when dealing with complex differential equations where traditional methods may fail.

For our equation, we assume a solution of the form:

\[ y = \sum_{n=0}^{\infty} a_n x^{n+r} \].

Here, \( a_n \) are the coefficients we need to determine, and \( r \) is a parameter that will be found using the indicial equation. By substituting this assumed solution back into the original differential equation, we turn the differential equation into an algebraic one, which can be solved step by step for the coefficients \( a_n \).

Indicial Equation

The indicial equation is crucial in the Frobenius method. It helps us find the possible values of the exponent \( r \) in our assumed series solution.

By substituting our series solution into the differential equation and simplifying, we form an equation from the lowest-order term (usually by setting the lowest power of x to zero). This equation is called the indicial equation:

\[ r(r-1) + 2r - 2 = 0 \].

Solving this quadratic equation gives us the possible values of \( r \), which are critical to finding the full series solution. In our example, we solve for more specific values of \( r \), yielding the exponents for our series.

Recurrence Relation

The recurrence relation helps us find the coefficients \( a_n \) in our series solution. Once we have the values of \( r \) from the indicial equation, we can substitute our series back into the differential equation and collect terms with similar powers of x.

This process yields a recurrence relation, which is an equation that relates each coefficient \( a_n \) to previous ones \( a_{n-1}, a_{n-2}, \dots \):

\[ a_{n+2} = f(a_n, a_{n-1}, \dots) \].

Using this relation, we can iteratively determine the coefficients starting from an initial term, often \( a_0 \) or \( a_1 \). For example, if \( r = 0 \) and our initial coefficient \( a_0 = 1 \), we can determine the next coefficients in the sequence to build our final series solution.