Question

Find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients. $$ 4 x^{2} y^{\prime \prime}+4 x y^{\prime}-(9-x) y=0 $$

Short answer

Question: Determine a fundamental set of Frobenius solutions and provide explicit formulas for the coefficients for the given differential equation: $$4x^2y'' + 4xy' - (9-x)y = 0$$ Answer: The fundamental set of Frobenius solutions with explicit formulas for the coefficients is given by: $$ y(x) = C_1 \sum_{n=0}^{\infty} (\prod_{k=0}^{n-1}{\frac{1 - 8k}{(2k+1)(2k+2)}})a_0 x^{2n} + C_2 \sum_{n=0}^{\infty} (\prod_{k=0}^{n-1}{\frac{1 - 8k-4}{(2k+1)(2k+3)}})a_1 x^{2n+1} $$

Step-by-step solution

Write down the given differential equation

The given differential equation is: $$ 4x^2y'' + 4xy' - (9-x)y = 0 $$

Rewrite the equation in standard form

Divide the equation by \(4x^2\) to obtain the standard form: $$ y'' + \frac{1}{x}y' - \frac{9-x}{4x^2}y = 0 $$

Apply the Frobenius method

Assume a power-series solution of the form: $$ y(x) = \sum_{n=0}^{\infty} a_n x^{n+r} $$ Then, differentiate with respect to \(x\) twice to find the first and second derivatives: $$ y'(x) = \sum_{n=0}^{\infty} (n+r) a_n x^{n+r-1} $$ $$ y''(x) = \sum_{n=0}^{\infty} (n+r)(n+r-1) a_n x^{n+r-2} $$

Simplify and find the recurrence relation

Substitute the power-series solutions for \(y\), \(y'\), and \(y''\) into the standard differential equation and equate coefficients of the same powers of \(x\): $$ \sum_{n=0}^{\infty} \left[(n+r)(n+r-1)a_n x^{n+r-2} + \frac{1}{x}\sum_{n=0}^{\infty}(n+r)a_n x^{n+r-1} - \frac{9-x}{4x^2}\sum_{n=0}^{\infty} a_n x^{n+r} \right] = 0 $$ After simplification, the recurrence relation is obtained: $$ a_{n+2} = \frac{1 - 4n - 4r}{(n+1)(n+r+1)}a_n $$

Obtain a fundamental set of solutions using the recurrence relation

Applying the recurrence relation, we can determine the coefficients for all terms in the power series. Using \(r=0\) as an example: $$ a_{n+2} = \frac{1 - 4n}{(n+1)(n+1)}a_n $$ Using this relation, we find a sequence of coefficients that converge: $$ a_0, a_2, a_4, \dots $$ Similarly, we can find another linearly independent solution using another value of \(r\) (if required). For example, if \(r_1\) results in another linearly independent sequence of coefficients, then the sequence is: $$ a_1, a_3, a_5, \dots $$ Combining both linearly independent solutions results in the fundamental set of solutions: $$ y(x) = C_1 \sum_{n=0}^{\infty} a_{2n} x^{2n} + C_2 \sum_{n=0}^{\infty} a_{2n+1} x^{2n+1} $$

Write the explicit formulas for the coefficients

Using the recurrence relation, we can write the explicit formulas for the coefficients: For the even-indexed coefficients: $$ a_{2n} = (\prod_{k=0}^{n-1}{\frac{1 - 8k}{(2k+1)(2k+2)}})a_0 $$ For the odd-indexed coefficients: $$ a_{2n+1} = (\prod_{k=0}^{n-1}{\frac{1 - 8k-4}{(2k+1)(2k+3)}})a_1 $$ The fundamental set of Frobenius solutions with explicit formulas for the coefficients is given by: $$ y(x) = C_1 \sum_{n=0}^{\infty} (\prod_{k=0}^{n-1}{\frac{1 - 8k}{(2k+1)(2k+2)}})a_0 x^{2n} + C_2 \sum_{n=0}^{\infty} (\prod_{k=0}^{n-1}{\frac{1 - 8k-4}{(2k+1)(2k+3)}})a_1 x^{2n+1} $$

Key concepts

Differential Equations

A differential equation is an equation that involves an unknown function and its derivatives. These equations are crucial in mathematics as they are used to describe a wide range of phenomena, from physics and engineering to biology and economics. The specific differential equation we are working with here is: \[ 4x^2y'' + 4xy' - (9-x)y = 0 \] This is a second-order linear differential equation because it involves the second derivative of the unknown function \( y(x) \). Solving a differential equation typically involves finding the function \( y(x) \) that makes the equation true. In this case, we are aiming to find solutions using the Frobenius method, which is especially useful for handling equations where a regular power series solution isn't suitable. This method helps when solutions near a singular point, such as \( x = 0 \), are needed.

Power Series Solution

A power series solution is a way of expressing a function as an infinite sum of terms, which involves powers of the variable, like so: \[ y(x) = \sum_{n=0}^{\infty} a_n x^{n+r} \] In our case, this expression assumes that \( y(x) \) can be expanded around a certain point, which helps solve differential equations when simple exact solutions are not feasible. The coefficients \( a_n \) in the series are determined using specific methods like the Frobenius method we are discussing here. By assuming this form and substituting it into the differential equation, we can derive the coefficients through recursion.

Recurrence Relation

Recurrence relations are equations that express each element of a sequence as a function of the preceding elements. They are extremely useful in determining the coefficients of a series solution. For our problem, after substituting into the differential equation, the recurrence relation is vital for expressing each coefficient \( a_{n+2} \) in terms of a previous coefficient \( a_n \). For instance: \[ a_{n+2} = \frac{1 - 4n}{(n+1)(n+1)}a_n \] This recurrence lets us compute all coefficients, given initial values, thus building up the series. These relations simplify complex calculations, helping us iteratively find a function's values, which is indispensable in the Frobenius method.

Fundamental Set of Solutions

A fundamental set of solutions for a differential equation is a set of independent solutions that can be combined to represent the general solution of the equation. For our second-order differential equation, we need two linearly independent solutions to form a fundamental set. Through the Frobenius method, we found two such sequences: one with coefficients \( a_{2n} \) and the other with coefficients \( a_{2n+1} \), representing even and odd terms, respectively. These solutions are generally written as: \[ y(x) = C_1 \sum_{n=0}^{\infty} a_{2n} x^{2n} + C_2 \sum_{n=0}^{\infty} a_{2n+1} x^{2n+1} \] Here, \( C_1 \) and \( C_2 \) are arbitrary constants determined by initial conditions. This combination of solutions provides the complete solution to the differential equation, capturing all behavior described by the equation.