Problem 18 - Advanced Engineering Mathematics

Question

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about \(x=0\). Form the general solution on the interval \((0, \infty)\). $$ 2 x^{2} y^{\prime \prime}-x y^{\prime}+\left(x^{2}+1\right) y=0 $$

Step-by-Step Solution

Answer

The indicial roots are \(\frac{1}{2} \pm i\). Use Frobenius method to find two solutions. The general solution is: \(y(x) = C_1 y_1(x) + C_2 y_2(x)\).

Step 1: Identify and Rewrite the Differential Equation

The given equation is \(2 x^{2} y^{\prime \prime} - x y^{\prime} + (x^{2} + 1) y = 0\). Identify the singularity at \(x = 0\). Here, the equation is in standard form for testing regular singular points and comparison with the standard form of a linear differential equation near a singular point: \(x^2 y^{\prime \prime} + x p(x) y^{\prime} + q(x) y = 0\).

Step 2: Compute Indicial Polynomial and Roots

To find the indicial polynomial, first consider the expression for the indicial equation: \(r(r-1) + pr + q = 0\), where **p** and **q** are the lowest-order coefficients from the standard differential equation. Rewriting the differential equation, knowing \(p(x) = -\frac{1}{2}\) and \(q(x) = \frac{x^2 + 1}{2x^2}\), we find: \[r(r - 1) - \frac{1}{2}r + \frac{1}{2} = 0\]. Solving this gives the indicial polynomial: \[2r^2 - r + 1 = 0\] resulting in roots \(\frac{1}{2} \pm \sqrt{-1} = \frac{1}{2} \mp i\sqrt{1}\). The roots are complex and do not differ by an integer.

Step 3: Apply Frobenius Method - First Solution

Start with the first root, \(r_1 = \frac{1}{2} + i\). We seek a series solution of the form \(y(x) = \sum_{n=0}^{\infty} a_n x^{n + r}\). Substitute into the differential equation and collect terms by powers of \(x\). Solve the resulting recursion for each coefficient \(a_n\) to form the first series solution.

Step 4: Apply Frobenius Method - Second Solution

Using the second root, \(r_2 = \frac{1}{2} - i\), similarly apply the Frobenius method to obtain the second linearly independent solution. Since the roots are not integers apart, the second solution can be constructed in the same way as the first.

Step 5: Form the General Solution

The general solution is a linear combination of the two linearly independent solutions found in steps 3 and 4. It can be expressed as: \[y(x) = C_1 y_1(x) + C_2 y_2(x)\] where \(C_1\) and \(C_2\) are arbitrary constants and \(y_1(x)\), \(y_2(x)\) are the Frobenius solutions obtained.

Key Concepts

Indicial Polynomial

An indicial polynomial is a crucial concept when dealing with differential equations, especially when employing the Frobenius method. It helps to determine possible solutions near singular points.
The polynomial is derived from the differential equation's standard form, often starting with a transformation. In essence, the indicial polynomial comes from substituting a power series into the differential equation and finding a relationship between coefficients.
For example, consider an equation like \(x^2 y^{\prime\prime} + x P(x) y^{\prime} + Q(x) y = 0\). It can be expressed in terms of an indicial equation \(r(r-1) + pr + q = 0\), using the coefficients from the expansions for P(x) and Q(x). Solving this polynomial gives the indicial roots, crucial for further analysis. In the given exercise, this results in roots \(\frac{1}{2} \pm i\).

Regular Singular Point

A regular singular point is found where a differential equation's leading coefficient vanishes, creating a peculiar effect on solutions. This point allows for more complex solutions using Frobenius method.
To identify, consider the form \(x^2 y^{\prime\prime} + xy^{\prime} + y = 0\). Find if near some point x (like \(x = 0\)), the behavior changes as powers of x decrease, but derivatives come with factors increasing rapidly.
This behavior suggests that solutions behave like powers of x times a series around the singularity. Resolving near \(x = 0\) vertices your path for series solutions that converge best around regular singular points. It's crucial since different solution types (like power series) don’t suffice near such points.

Series Solutions

Series solutions offer a way to solve differential equations by transforming them into infinite sums. This method becomes invaluable, especially near singular points.
Here's why they're useful: not all differential equations yield to simple function solutions, but many can be approached by assuming solutions in series form.

• Start with the general series form: \(y(x) = \sum_{n=0}^{\infty} a_n x^{n + r}\), substituting into the differential equation.
• Arrange terms to reveal a pattern, connecting coefficients.
• Helps solve each coefficient recursively, ensuring correct terms add to zero.

For the given exercise, it's the Frobenius series that unlocks two independent solutions, adjusted by allowing complex roots from the indicial polynomial.

Differential Equations

Differential equations form the backbone of modeling real-world phenomena where change is continuous, such as physics, engineering, and economics.
These equations relate functions to their derivatives, explaining how variables evolve over time or space. Their solutions provide crucial insights into the underlying processes.

• Ordinary differential equations (ODEs) involve functions of one variable and their derivatives.
• Partial differential equations (PDEs) involve multiple variables.

In this exercise, we work with an ODE characterized by a mix of derivatives, whose coefficients change, implying varying rates inside the equation. An equation like \(2 x^{2} y^{\prime \prime} - x y^{\prime} + (x^{2} + 1) y = 0\) exemplifies such, with distinctive coefficients adding complexity, and illustrating how tough solutions might be without series approaches.