Question

Solve the following differential equations by power series. $$ y^{\prime \prime}-4 x y^{\prime}+\left(4 x^{2}-2\right) y=0 $$

Short answer

The solution is a power series where coefficients are determined by the recurrence relation \(a_{n+2} = \frac{4(n+1)a_{n+1} - 2a_n + 4a_{n-2}}{(n+2)(n+1)}\). Initial terms are defined using \(a_0\) and \(a_1\).

Step-by-step solution

Assume a Power Series Solution

Assume the solution is of the form: \( y(x) = \sum_{n=0}^{fty}a_n x^n \). Compute the first and second derivatives: \( y'(x) = \sum_{n=1}^{fty}na_n x^{n-1} \) \( y''(x) = \sum_{n=2}^{fty}n(n-1)a_n x^{n-2} \).

Substitute Series into the Differential Equation

Substitute the series expressions into the differential equation \( y'' - 4xy' + (4x^2 - 2)y = 0 \): \( \sum_{n=2}^{fty}n(n-1)a_n x^{n-2} - 4x \sum_{n=1}^{fty}na_n x^{n-1} + (4x^2 - 2) \sum_{n=0}^{fty}a_n x^n = 0 \).

Reindex the Series

Reindex all sums to have the same power of x: \( \sum_{n=0}^{fty}(n+2)(n+1)a_{n+2} x^n - 4 \sum_{n=0}^{fty}(n+1)a_{n+1} x^n + 4 \sum_{n=0}^{fty}a_n x^{n+2} - 2 \sum_{n=0}^{fty}a_n x^n = 0 \).

Combine Like Terms

Combine the series terms so that coefficients can be compared at each power of x: \( (2a_2 - 2a_0) + \sum_{n=1}^{fty}[(n+2)(n+1)a_{n+2} - 4(n+1)a_{n+1} - 2a_n]x^n + 4 \sum_{n=0}^{fty}a_n x^{n+2} = 0 \).

Set Up Equation for Recurrence Relation

Since the series must equal zero for all x, set the coefficients of each power of x to zero: For \(x^0\): \(2a_2 - 2a_0 = 0\) giving \(a_2 = a_0\). For \(x^n\): \[(n+2)(n+1)a_{n+2} - 4(n+1)a_{n+1} + 4a_{n-2} - 2a_n = 0\].

Solve the Recurrence Relation

Solve for \(a_{n+2}\) in terms of the previous coefficients: \[(n+2)(n+1)a_{n+2} = 4(n+1)a_{n+1} - 2a_n - 4a_{n-2}\]. Divide by \((n+2)(n+1)\): \[a_{n+2} = \frac{4(n+1)a_{n+1} - 2a_n + 4a_{n-2}}{(n+2)(n+1)}\].

Determine Initial Conditions

Use initial conditions to solve for specific coefficients: Let \(a_0\) and \(a_1\) be constants. Thus, the recurrence relation solution is as follows: \(a_2 = a_0\) \(a_3\) can be found using further terms based on initial values of \(a_0\) and \(a_1\).

Key concepts

recurrence relation

A recurrence relation establishes a connection between various terms in a sequence. In the context of solving differential equations using power series, we derive a recurrence relation to express higher-order coefficients in terms of lower-order ones. For our given problem:
The recurrence relation links the coefficient of the term with power \(x^{n+2}\) to the coefficients of the terms with powers \(x^n, x^{n-1}, \text{and} x^{n-2}\). Specifically, we found:
a_{n+2} = \frac{4(n+1)a_{n+1} - 2a_n + 4a_{n-2}}{(n+2)(n+1)}\.
The next step is to solve this relation to determine the coefficients starting from the initial conditions.

initial conditions

Initial conditions refer to the values of the function and its derivatives at a particular point, often at \(x=0\). These values are crucial for determining the constants in the power series solution of a differential equation.
Let \(a_0\) and \(a_1\) be constants representing the initial conditions such as the initial value of the function and its first derivative. For example, we assume:

• \(y(0) = a_0\)
• \(y'(0) = a_1\)

For our specific problem, these initial conditions help determine the specific values of the coefficients in the power series.

power series solution

A power series solution approximates a function around a particular point, typically \(x=0\), by expressing it as an infinite sum of terms. For the differential equation \( y^{\backslashprime\backslashprime} - 4xy^{\backslashprime} + (4x^2 - 2)y = 0 \), we assume a solution of the form
\( y(x) = \backslashsum_{n=0}^{\fty}a_n x^n\).Substituting this assumed solution into the differential equation allows us to equate coefficients of like powers of \(x\), which eventually leads to a recurrence relation.
This series enables us to express the solution as:
a_{0} \backslash+ a_{1}x \backslashplus \backslashldots\.

second-order differential equations

Second-order differential equations involve the second derivative of the unknown function. For our equation \( y^{\backslashprime\backslashprime} - 4xy^{\backslashprime} + (4x^2 - 2)y = 0\), the solution process involves:

• Assuming a power series form for y(x).
• Finding power series for the first and second derivatives.
• Substituting these series into the original equation.
• Reindexing and combining like terms to align all powers of \(x\).

This method transforms a differential equation into an algebraic relationship between coefficients. The resulting power series solution can approximate the behavior of the original function.