What is the Series Solution for the Given Equation: x^2 y xy' - xy 0?

When solving differential equations, one useful and powerful method is the series solution method. This article will guide you step-by-step through the process for solving the differential equation x^2 y xy' - xy 0. Let's explore how to derive the power series solution to this equation and the recurrence relation for the coefficients.

1. Assumption of a Power Series Solution

First, we assume a power series solution for the function y(x) in the form:

y(x) sum_{n0}^{infty} a_n x^n

where the coefficients a_n need to be determined.

2. Derivatives of the Power Series

Next, we find the derivatives of y(x):

y'(x) sum_{n1}^{infty} n a_n x^{n-1} y''(x) sum_{n2}^{infty} n(n-1) a_n x^{n-2}

Now, let's substitute these into the differential equation step by step.

3. Substitution and Simplification

First, consider the term x^2 y(x) and substitute the power series:

x^2 y(x) x^2 sum_{n0}^{infty} a_n x^n sum_{n0}^{infty} a_n x^{n 2}

To align the powers of x, shift the index by letting n 2m:

x^2 y(x) sum_{m2}^{infty} a_{m-2} x^m sum_{n0}^{infty} a_n x^{n 2}

Next, consider the term xy'(x) and substitute the derivative:

xy'(x) x sum_{n1}^{infty} n a_n x^{n-1} sum_{n1}^{infty} n a_n x^n

And the term -xy(x) and substitute the original series:

-xy(x) -x sum_{n0}^{infty} a_n x^n sum_{n0}^{infty} -a_n x^{n 1}

To combine these, we need all terms expressed in terms of x^n. Let's combine the equations:

sum_{n0}^{infty} a_n x^{n 2} sum_{n1}^{infty} n a_n x^n - sum_{n0}^{infty} a_n x^{n 1} 0

Now, let's adjust the indices to align:

sum_{n0}^{infty} a_n x^{n 2} sum_{n1}^{infty} n a_n x^n - sum_{n1}^{infty} a_{n-1} x^n 0

For the first term, let n 2 m:

sum_{m2}^{infty} a_{m-2} x^m sum_{n0}^{infty} a_n x^{n 2}

Thus, rewriting all sums in terms of x^n:

sum_{n0}^{infty} a_n x^{n 2} sum_{n1}^{infty} n a_n x^n - sum_{n1}^{infty} a_{n-1} x^n 0

Combine the second sum:

sum_{n0}^{infty} a_n (n 2) x^n - sum_{n0}^{infty} a_{n 1} x^n 0

4. Deriving the Recurrence Relation

From the combined equation, we can equate the coefficients of x^n to zero:

(n 2) a_n - a_{n 1} 0

Solving for a_{n 1} gives:

a_{n 1} (n 2) a_n

5. Iterative Calculation of Coefficients

This recurrence relation allows us to compute the coefficients iteratively once initial conditions a_0 and a_1 are given. The solution is then:

y(x) a_0 a_1 x a_2 x^2 a_3 x^3 cdots

where:

a_2 2 a_1 a_3 3 a_2 3 (2 a_1) 6 a_1 a_4 4 a_3 4 (6 a_1) 24 a_1

and so on. This process forms the power series solution.

Summary

In summary, the series solution involves:

Assuming a power series solution for y(x). Calculating derivatives and substituting them into the differential equation. Collecting like terms to derive a recurrence relation for the coefficients.

The series solution depends on the values of a_0 and a_1, which can be determined from initial or boundary conditions if provided.