Problem 23 - Advanced Engineering Mathematics

Question

Find two power series solutions of the given differential equation about the ordinary point \(x=0\). $$ (x-1) y^{\prime \prime}+y^{\prime}=0 $$

Step-by-Step Solution

Answer

The power series solutions are \(y_1(x) = C_1\) and \(y_2(x) = C_2x\).

Step 1: Identify the Point of Expansion

The differential equation is given as \((x-1) y^{\prime \prime}+y^{\prime}=0\). Since we need the power series solution about \(x=0\), it's important to note that \(x=0\) is an ordinary point of the differential equation.

Step 2: Assume a Power Series Solution

Assume a power series solution of the form \(y = \sum_{n=0}^{\infty} a_n x^n\). Differentiate this series to find \(y'\) and \(y''\):\[y' = \sum_{n=1}^{\infty} n a_n x^{n-1}\]\[y'' = \sum_{n=2}^{\infty} n (n-1) a_n x^{n-2}\]

Step 3: Substitute and Simplify

Substitute the expressions for \(y'\) and \(y''\) into the differential equation. Since the equation is \((x-1) y^{\prime \prime}+y^{\prime}=0\), plug in \(y''\) and \(y'\):\[(x-1) \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + \sum_{n=1}^{\infty} n a_n x^{n-1} = 0\]Multiply out the terms in the first sum:\[\sum_{n=2}^{\infty} n(n-1) a_n x^{n-1} - \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} + \sum_{n=1}^{\infty} n a_n x^{n-1} = 0\]

Step 4: Combine and Rearrange Series Terms

Combine and rearrange terms in the series to have a common index. This involves adjusting the indices so they all start from the same index and then combine terms:\[n=0: \, (a_1) + \sum_{n=2}^{\infty} \left[n^2 a_n x^{n-1} - n(n-1) a_n x^{n-2}\right] = 0\]

Step 5: Solve for Recurrence Relation

From the setup in Step 4, find a pattern or relation between the coefficients. Equate the powers of \(x\) to zero to get a recurrence relation:\[a_1 = 0\]\[n^2 a_n x^{n-1} = (n(n-1) a_n - (n-1) n a_n)x^{n-2}\]Since terms for different powers of \(x\) must independently be zero, solve for the relation of coefficients.

Step 6: Derive the Solution

Take the free terms left after solving the recurrence relation and form the series. Consider the initial conditions:\(a_1 = 0\) and then derive other coefficients \(a_n\) assuming some starting conditions like \(a_0 = 1\) and \(a_0 = 0\). The solutions will then be of the form:For the first solution:\[ y_1 = a_0 + 0 + ... = a_0 + 0 \, \text{as other terms vanish}...\]For the second solution:Slightly modify the conditions such as setting vectorial conditions to derive other solutions.

Step 7: Present Power Series Solutions

After deriving the coefficients for different conditions, write the solutions in terms of power series:\[ y_1(x) = C_1\] for the solution derived by initial \(a_0 = 1\).\[ y_2(x) = C_2x\] for the solution assuming start at \(a_1\) or other.

Key Concepts

Differential Equation

A differential equation is an equation that involves the derivatives of a function. These equations are at the heart of calculus and are used to describe various phenomena in science and engineering. For instance, they model how heat spreads in a metal rod or how populations of organisms grow over time.

• Types: Differential equations can be ordinary (ODE) or partial (PDE). Ordinary differential equations, like the one in our problem, involve functions of a single variable and their derivatives.
• Components: They often stem from physical laws where one relates rates of change (derivatives) to quantities (functions).

In our example, we have the differential equation \((x-1) y^{\prime \prime} + y^{\prime} = 0\), which is an ODE since it involves derivatives of one variable, \(x\).
To solve this, we use series solutions, particularly focusing on constructing a series expansion that accounts for all conditions specified in the equation.

Ordinary Point

When dealing with differential equations, it's essential to understand the term "ordinary point." An ordinary point is a value of \(x\) where the functions multiplying the highest derivative in the differential equation do not make the solution undefined or complicated. Essentially, it is a point where the solution behaves nicely.

• Identification: To identify an ordinary point, check where the coefficients of the highest derivatives become zero. If they are non-zero at a point, it's ordinary.
• Implication: At ordinary points, solutions can be determined using power series representations, which simplifies solving differential equations considerably.

In our exercise, \(x=0\) is an ordinary point because the coefficient of \(y^{\prime\prime}\) is \(x-1\), and it does not cause any issues at \(x=0\). This allows us to use power series solutions about \(x=0\), ensuring that the series converges properly, and solutions can be found smoothly.

Recurrence Relation

A recurrence relation is like a recipe that gives you the ingredients (terms of a series) needed to produce a sequence based upon previously known elements. It's a powerful tool that helps in determining the terms of a sequence or series systematically by relying on relationships between consecutive terms.

• Purpose: In the context of differential equations, recurrence relations help in finding the coefficients of a power series solution. It simplifies the series construction since each term only relies on its predecessors.
• Implementing: Once the series for \(y\), \(y'\), and \(y''\) are substituted in the differential equation, combining and comparing terms gives a recurrence relation.

In our differential equation, we derived a recurrence relation through the substitution and simplification process, which allowed us to express each coefficient in terms of the previous ones. Solving these relations systematically helps construct the power series solution to the differential equation.