Question

Find two linearly independent solutions of the Bessel equation of order \(\frac{3}{2}\), $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{9}{4}\right) y=0, \quad x>0 $$

Short answer

The two linearly independent solutions for the Bessel equation of order \(3/2\) are: - \(y_1(x) = \sum_{n=0}^{\infty} a_n x^{n+\frac{1}{2}}\) - \(y_2(x) = \sum_{n=0}^{\infty} a_n x^{n-1}\)

Step-by-step solution

Identify the terms and expressions in the Bessel equation

The given Bessel equation of order \(3/2\) is: $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{9}{4}\right) y=0, \quad x>0 $$ Here, \(y(x)\) is the unknown function that we want to find, and \(y'\) and \(y''\) represent first and second derivatives of \(y(x)\) with respect to \(x\).

Assume a power series solution

We will use the Frobenius method to find a power series solution to the Bessel equation. Let's assume a power series solution of the form: $$ y(x) = \sum_{n=0}^{\infty} a_n x^{n + s} $$ Where \(a_n\) are coefficients, and \(s\) is an unknown constant.

Calculate the derivatives

Next, we'll need to find the first and second derivatives of \(y(x)\). First derivative: $$ y'(x) = \sum_{n=0}^{\infty} (n+s) a_n x^{n+s-1} $$ Second derivative: $$ y''(x) = \sum_{n=0}^{\infty} (n+s)(n+s-1) a_n x^{n+s-2} $$

Substitute derivatives back into the Bessel equation

Now, substitute the power series for \(y(x)\) and its derivatives back into the Bessel equation: $$ x^2 \sum_{n=0}^{\infty} (n+s)(n+s-1) a_n x^{n+s-2} + x \sum_{n=0}^{\infty} (n+s) a_n x^{n+s-1} + \left( x^{2} - \frac{9}{4} \right) \sum_{n=0}^{\infty} a_n x^{n+s} = 0 $$

Combine the summations and simplify the equation

Let's combine the summations of the terms and simplify the equation: $$ \sum_{n=0}^{\infty} a_n x^{n+s} \left[ (n+s)(n+s-1) + (n+s) - \frac{9}{4} \right]= 0 $$ The above equation will only be identically true if the expression inside the square brackets is equal to zero for each \(n\). So, $$ (n+s)(n+s-1) + (n+s) - \frac{9}{4} = 0 $$

Find the values for the constant 's'

Solve this quadratic equation for 's': $$ (n+s)(n+s-1) + (n+s) - \frac{9}{4} = 0 $$ The quadratic equation has two possible solutions for 's': $$ s_1 = \frac{1}{2}, \quad s_2 = -1 $$

Obtain the power series solutions

Substituting the values of 's' in the power series solution, we will get two linearly independent solutions for the Bessel function: $$ y_1(x) = \sum_{n=0}^{\infty} a_n x^{n+\frac{1}{2}} $$ $$ y_2(x) = \sum_{n=0}^{\infty} a_n x^{n-1} $$ These two power series solutions \(y_1(x)\) and \(y_2(x)\) are linearly independent solutions of the given Bessel equation of order \(\frac{3}{2}\).

Key concepts

Linearly Independent Solutions

When dealing with differential equations, like the Bessel equation, it's crucial to find solutions that are linearly independent. But what does this mean? In simple terms, solutions are linearly independent if no solution can be written as a combination of the others. For example, if you can express one solution as a multiple, or sum, of another, they are not independent.

Linear independence is important because it helps us build the general solution of differential equations. We can't fully describe all potential solutions without having a set of linearly independent ones. It ensures we cover all possible behaviors the system might exhibit.

• *Definition:* Solutions are linearly independent if their Wronskian (a special determinant related to them) is non-zero.
• *Example:* In our Bessel equation case, the functions \(y_1(x)\) and \(y_2(x)\) are linearly independent. This means they can describe all solutions of the given equation.

Understanding linearly independent solutions gives us a complete toolkit to solve and analyze differential equations effectively in mathematics and science.

Frobenius Method

The Frobenius Method is a powerful tool used to find solutions to differential equations like the Bessel equation, particularly when the equation does not have solutions in simple algebraic forms.

Essentially, the method involves expanding a solution in a series called a power series. But unlike a regular power series, which starts with \(x^0\), the Frobenius series allows for solutions that look like \(x^s\) where \(s\) is a constant that might not be an integer. This added flexibility is crucial when regular series solutions fail.

• *Steps Involved:* We guess a series solution \(y(x) = \sum_{n=0}^{\infty} a_n x^{n+s}\). Then, we differentiate this series to find \(y'(x)\) and \(y''(x)\), substitute these back into the original equation, and solve for the coefficients \(a_n\) and the constant \(s\).
• *Benefits:* The method allows finding all possible solutions at once, especially in cases involving singular points not solvable by simpler methods.

This method is named after the mathematician Frobenius who developed this approach to better tackle complex differential equations with singularities.

Power Series Solution

A power series solution is a common method for solving differential equations. It involves expressing the solution as a series of powers of \(x\). This is especially useful when the equation doesn't yield to simpler methods like separation of variables or integrating directly.

For the Bessel equation, the power series takes the form \(y(x) = \sum_{n=0}^{\infty} a_n x^{n+s}\). Each term in the series has a coefficient \(a_n\) and is raised to the power \(n+s\). This approach turns solving the differential equation into finding a pattern for the coefficients. If we can determine all \(a_n\)'s, we essentially have the solution.

• *Simplifying Benefits:* Instead of dealing directly with complex functions, we break them down into a sum of simpler components.
• *Flexibility:* The range, \(s\), allows adapting the series to initial conditions or boundary conditions present in specific problems.

Using power series solutions is not only about finding a solution but also gaining deeper insights into the behavior and properties of the solution near particular values of \(x\). It is a fundamental technique in both pure and applied mathematics.