Question

Solve the following differential equations by series and also by an elementary method and verify that your solutions agree. Note that the goal of these problems is not to get the answer (that's easy by computer or by hand) but to become familiar with the method of series solutions which we will be using later. Check your results by computer. $$y^{\prime \prime}-4 x y^{\prime}+\left(4 x^{2}-2\right) y=0$$

Short answer

Assume a series solution, derive the recurrence relation, then solve the differential equation by an elementary method and compare both solutions.

Step-by-step solution

- Assume Series Solution

Assume a solution of the form \[ y = \sum_{n=0}^{\infty} a_n x^n \]

- Find First and Second Derivatives

Calculate \[ y' = \sum_{n=1}^{\infty} a_n n x^{n-1} \]and \[ y'' = \sum_{n=2}^{\infty} a_n n (n-1) x^{n-2} \].

- Substitute into ODE

Substitute the series forms of \( y \), \( y' \), and \( y'' \) into the differential equation \[ y'' - 4xy' + (4x^2 - 2)y = 0 \].

- Equate Coefficients

Combine and compare coefficients of like powers of \( x \) to obtain a recurrence relation for the coefficients \( a_n \). Solve the recurrence relation to determine the series terms.

- Find Elementary Solution

Solve the differential equation using an elementary method, such as the method of undetermined coefficients or variation of parameters.

- Compare Solutions

Compare the solution obtained from the series method with the one obtained from the elementary method to verify that they match.

- Verification by Computer

Using a mathematical software or computer algebra system (like WolframAlpha, MATLAB, or Mathematica), check if both solutions satisfy the original differential equation.

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations involving a function and its derivatives. In our exercise, the ODE given is of the second order, as it involves the second derivative of the function y(x). The general form of a second-order ODE is typically written as: f(x, y, y', y'') = 0.
A common example is Newton's second law of motion. Solving an ODE means finding a function y(x) that satisfies the equation within a certain domain. Methods like separation of variables, integrating factors, or using power series can help find solutions.

Power Series

A power series is a series of the form: y = Σ n=0^ ∞. a_.n x^.n
In step 1 of our solution, we assume y(x) can be expressed as a power series. This helps us convert the differential equation into an algebraic form. We calculate the derivatives, which simplifies substitution back into the original ODE. By comparing the coefficients of different powers of x, we get a recurrence relation to find the coefficients a_.n_.
Power series expansions are powerful tools for approximating solutions near a point.

Recurrence Relation

A recurrence relation is an equation that defines a sequence based on previous terms. In our problem, after substituting the power series into the ODE (Step 3), we compare the coefficients of like powers of x. This gives us a recurrence relation for the coefficients a_.n: a_.n = n(n-1) a_n - 4 n x^(n-1) a_n + (4 x^2 - 2) a_n = 0.
Solving this recurrence helps us find the coefficients needed to build the power series solution for y(x). Recurrence relations are essential for understanding the structure of solutions.

Elementary Methods of Solving Differential Equations

Elementary methods for solving differential equations include techniques like separation of variables, integrating factors, undetermined coefficients, and variation of parameters. In step 5, we use such an elementary method to solve our ODE.
For instance, the method of undetermined coefficients involves guessing a solution form and determining the coefficients that satisfy the ODE.
By solving with an elementary method and comparing it to the series solution, we ensure our solutions are correct and confirm our understanding of both approaches.
These methods are useful in solving differential equations analytically, providing clear insights into the behavior of the system.