Problem 32 - Advanced Engineering Mathematics

Question

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the recurrence relation found by the method of Frobenius first with the largest root \(r_{1}\). How many solutions did you find? Next use the recurrence relation with the smaller root \(r_{2}\). How many solutions did you find? $$ x(x-1) y^{\prime \prime}+3 y^{\prime}-2 y=0 $$

Step-by-Step Solution

Answer

One power series solution from the largest root is found; no new solution from the smaller root.

Step 1: Identify Regular Singular Points

The differential equation is given as \(x(x-1) y'' + 3y' - 2y = 0\). We can determine the regular singular points by examining where the coefficients would become undefined when in standard form (with \(x^2\) as the leading coefficient). Here, the singular points are where \(x = 0\) and \(x = 1\) appear due to the factor \(x(x-1)\).

Step 2: Determine Indicial Equation

Using the method of Frobenius, the solution around the singular point \(x=0\) is \(y = x^r \sum_{n=0}^{abla} a_n x^n\). Substitute this into the differential equation and analyze the coefficients of powers of \(x\) to form the indicial equation. This process yields the indicial equation: \(r(r-1) + 3r = 0\), simplifying to \(r^2 + 2r = 0\).

Step 3: Solve Indicial Equation

Solving the indicial equation \(r^2 + 2r = 0\) yields the roots \(r_1 = 0\) and \(r_2 = -2\). These roots differ by an integer, confirming the problem statement.

Step 4: Use Frobenius Method with Largest Root \(r_1 = 0\)

Assume the solution \(y = \sum_{n=0}^{abla} a_n x^n\). Substitute this form in the differential equation and derive the recurrence relation for coefficients \(a_n\). The recurrence relation allows us to determine the coefficients successively, providing the solutions.

Step 5: Analyze Solutions with Largest Root

Using the largest root \(r_1 = 0\), we find that one distinct solution emerges from the recurrence relation, which can be expressed as a power series.

Step 6: Use Frobenius Method with Smaller Root \(r_2 = -2\)

Now assume \(y = x^{-2} \sum_{n=0}^{abla} a_n x^n\). Substituting this into the equation produces a different recurrence relation for the coefficients \(a_n\).

Step 7: Analyze Solutions with Smaller Root

For \(r_2 = -2\), we attempt to construct a second solution. However, this root does not provide an additional independent solution due to the integer difference with the first root; thus, it does not yield a new solution.

Step 8: Conclude on Number of Solutions

Ultimately, the method yields one independent power series solution based on the largest root and no additional independent solution due to the roots' integer difference.

Key Concepts

Regular Singular Point

A regular singular point in differential equations is where the equation's coefficients become undefined or infinite yet maintain a certain type of behavior. In the given differential equation, we identify points by inspecting the leading term's coefficient. The differential equation is \( x(x-1) y'' + 3y' - 2y = 0 \). Here, the presence of \(x(x-1)\) implies singular points at \(x = 0\) and \(x = 1\).

A point is considered regular singular if any solutions behave predictably near the singularity. This allows special techniques like the Frobenius method to be applied, giving us valuable insight into the nature of the solutions near these points. Thus, recognizing \(x = 0\) as a regular singular point qualifies us to apply these methods to find the indicial equation and attempt to derive the series solutions.

Indicial Equation

The indicial equation is a key component when using the Frobenius method for solving differential equations. It arises from substituting a power series solution into the differential equation and requiring that the lowest power of the series has a coefficient of zero.

For the differential equation at hand, surrounding the regular singular point \(x=0\), we propose a solution form, \(y = x^r \sum_{n=0}^{\infty} a_n x^n \). Substituting this series into the equation and equating the lowest power of \(x\) gives us the indicial equation.

In this case, the indicial equation comes out to be \(r(r-1) + 3r = 0\), which simplifies to \(r^2 + 2r = 0\). Solving this yields roots \(r_1 = 0\) and \(r_2 = -2\). The integer difference between these roots plays a critical role in determining the number and type of solutions.

Differential Equation Solutions

Solutions to differential equations, especially those around regular singular points, are often sought in power series form. This is due to their ability to closely approximate functions near singularities, which are points where coefficients of the differential equation might otherwise be undefined.

By using the largest root from the indicial equation, \(r_1 = 0\), one solution can be developed through a series expansion. For the differential equation \(x(x-1)y'' + 3y' - 2y = 0\), the solution derived from \(r_1 = 0\) leads to a unique power series solution.

However, when considering the smaller root \(r_2 = -2\), constructing a new solution becomes challenging because of the roots' specific difference of 2. Typically, large differences between roots allow independent solutions, but here, the integer difference obstructs finding a second independent solution, resulting in just one valid series solution.

Recurrence Relation

The recurrence relation is a mathematical expression derived during the application of the Frobenius method, defining how each coefficient in a power series solution relates to its preceding ones. This relationship helps in determining all the coefficients of the series.

To use the recurrence relation, substitute a proposed series form like \(y = \sum_{n=0}^{\infty} a_n x^n\) into the differential equation. Then match coefficients for corresponding powers of \(x\) to create equations for \(a_n\) in terms of previous coefficients. Such relation is pivotal for calculating specific terms of the series, thus defining the function solution accurately.

For our equation, the recurrence relation comes into effect especially for the largest root \(r_1 = 0\). It allows us to solve sequentially for coefficients \(a_n\), leading to a complete power series solution for the differential equation near \(x=0\). With the smaller root \(r_2 = -2\), no new independent series solution arises, aligning with the integer difference theory of indicial roots.