Problem 9 - Advanced Engineering Mathematics

Question

Use an appropriate infinite series method about \(x=0\) to find two solutions of the given differential equation. $$ 2 x y^{\prime \prime}+y^{\prime}+y=0 $$

Step-by-Step Solution

Answer

The solutions are power series: \( y_1 = c_0 (1 - \frac{x^2}{3} + \frac{x^4}{30} + \ldots) \) and \( y_2 = c_1 (x - \frac{x^3}{6} + \frac{x^5}{120} + \ldots) \).

Step 1: Set up the power series

Assume a solution of the form of a power series around \( x = 0 \): \[ y(x) = \sum_{n=0}^{fty} a_n x^n \] Find its first and second derivatives:\[ y'(x) = \sum_{n=1}^{fty} n a_n x^{n-1} \]\[ y''(x) = \sum_{n=2}^{fty} n(n-1) a_n x^{n-2} \]

Step 2: Substitute into the differential equation

Substitute \( y = \sum a_n x^n \), \( y' = \sum n a_n x^{n-1} \), and \( y'' = \sum n(n-1) a_n x^{n-2} \) into the given differential equation:\[ 2x\left(\sum_{n=2}^{fty} n(n-1) a_n x^{n-2}\right) + \sum_{n=1}^{fty} n a_n x^{n-1} + \sum_{n=0}^{fty} a_n x^n = 0 \]

Step 3: Align and combine the series

Adjust indices and align powers of \( x \) in all sums. Rewrite:\[ 2\sum_{n=2}^{fty} n(n-1) a_n x^{n-1} + \sum_{n=1}^{fty} n a_n x^{n-1} + \sum_{n=0}^{fty} a_n x^n = 0 \]Combine like terms and adjust indices so that all summations begin at the same power. This will facilitate factorization of the same power of \( x \).

Step 4: Factor and solve coefficient equations

To cancel each power of \( x \), their overall coefficient must equate to zero. For \( n \geq 2 \):\[ 2n(n-1)a_n + n a_n + a_{n-2} = 0 \] Which simplifies to recursion:\[ a_n = -\frac{a_{n-2}}{2n^2 + n} \] For the first few terms, find specific values using initial conditions like \( a_0 \) and \( a_1 \).

Step 5: Determine initial conditions and solve

Set \( a_0 = c_0 \) and \( a_1 = c_1 \) for arbitrary constants \( c_0 \) and \( c_1 \). Using initial conditions, plug in and solve using the recursion formula to get:\[ y_1 = c_0 (1 - \frac{x^2}{3} + \frac{x^4}{30} + \ldots ) \] \[ y_2 = c_1 (x - \frac{x^3}{6} + \frac{x^5}{120} + \ldots ) \] This represents two linearly independent solutions as power series.

Key Concepts

Differential Equations

Differential equations are equations that involve functions and their derivatives. They are used to describe various phenomena such as motion, growth, and decay. A differential equation typically represents a relationship between the rates at which things change. For example, in physics, differential equations can describe the behavior of moving objects or heat flow.
A differential equation can be ordinary or partial. The ordinary differential equations (ODEs) involve functions of a single variable and their derivatives, while partial differential equations (PDEs) involve multiple variables and their partial derivatives.
Solving differential equations often requires finding a unique function or a set of functions that satisfy the equation. The solutions can be expressed in different forms, such as explicitly with functions or implicitly with power series. In this exercise, we focus on solving the differential equation using power series methods.

Recursion Relations

Recursion relations are equations that define sequences based on preceding terms. They are an essential tool in solving differential equations when using a power series approach. Recursion allows us to express higher-order terms in a series using earlier terms. This method simplifies the complex problem of finding the entire series solution.
In the context of differential equations, we substitute a power series into the equation and align the terms based on their powers of the variable. This alignment results in coefficients linked by recursion relations that must equate to zero for the equation to hold true.
As demonstrated, by solving these recursion relations, we can find specific coefficients for each term in the power series. This method provides insight into the kinds of solutions the differential equation may have, often leading to discovering infinite series solutions.

Infinite Series Method

The infinite series method is a powerful technique for solving differential equations. The idea is to assume a solution in the form of an infinite series and use this assumption to transform the differential equation into a set of simpler algebraic equations.
The process begins by expressing the solution as an infinite power series, as demonstrated in the step-by-step solution. We then find the derivatives of this series, substitute them into the original differential equation, and solve for the coefficients by equating powers of the variable.
This method is exceptionally helpful for differential equations that are difficult to solve using standard techniques. It provides approximate solutions that can be tailored to desired accuracy by adjusting the number of terms in the series. The series solutions often reveal important properties about the behavior of solutions, especially near singular points where other methods might fail.