Question

Determine the general solution of the given differential equation that is valid in any interval not including the singular point. \(x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=0\)

Short answer

#Short Answer# To find the general solution of the given second-order homogeneous differential equation with variable coefficients, we assume a solution in the form of a power series and compute its derivatives. Plugging the power series and its derivatives into the differential equation, we match the powers of x to obtain a recurrence relation for the coefficients. Using this recurrence relation, we find the general solution in terms of a power series, express the first few terms, and identify patterns or simplifications. Finally, we write down the more generalized terms in the series and combine them to obtain the desired general solution.

Step-by-step solution

Assume a solution in the form of a power series

Let's assume a solution to the differential equation in the form of a power series: $$y(x) = \sum_{n=0}^{\infty} a_n x^n$$

Compute the derivatives of y(x)

We differentiate y(x) twice with respect to x to obtain the first and second derivatives: $$y'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1}$$ $$y''(x) = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}$$

Substitute the power series and its derivatives into the given differential equation

Plug the power series of y(x), y'(x), and y''(x) into the given differential equation: $$x^2 \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2} - 3x \sum_{n=1}^{\infty} n a_n x^{n-1} + 4 \sum_{n=0}^{\infty} a_n x^n = 0$$

Match the powers of x

In order for both sides of the equation to be equal, the coefficients of the same powers of x must be equal. To match the powers, we multiply the sums by the appropriate powers of x, and then change the indices to have the same lower limit: $$\sum_{n=0}^{\infty} [(n+2)(n+1) a_{n+2} - 3(n+1) a_{n+1} + 4 a_n] x^n = 0$$

Determine the recurrence relation

Since both sides of the equation must be equal, the term inside the square brackets must be zero for every n: $$(n+2)(n+1) a_{n+2} - 3(n+1) a_{n+1} + 4 a_n = 0$$ Now, we can express the coefficients \(a_{n+2}\) in terms of \(a_{n+1}\) and \(a_n\) using the recurrence relation: $$a_{n+2} = \frac{3(n+1)}{(n+2)(n+1)-4} a_{n+1} - \frac{4}{(n+2)(n+1)-4} a_n$$

Find the general solution using the coefficients from the recurrence relation

The general solution y(x) can now be expressed using the coefficients found from the recurrence relation: $$y(x) = \sum_{n=0}^{\infty} a_n x^n$$ Since we have an expression for \(a_{n+2}\) in terms of \(a_{n+1}\) and \(a_n\), we can find the first few terms of the series and notice any possible patterns or simplifications. Then we can write down the more generalized terms in the series and combine them to form the desired general solution.

Key concepts

General Solution

Understanding the general solution of a differential equation is crucial for comprehending its behavior and predicting the system it describes. In the context of the given exercise, the general solution represents all possible functions y(x) that satisfy the differential equation throughout a specified domain, excluding singular points. A singular point is essentially where the mathematical behavior of the equation changes, like where a coefficient function becomes zero.

In our situation, the general solution is not a single function but rather a series of functions expressed as a power series. The power series is infinitely long, which can precisely capture the complex nature of solutions to such differential equations. This approach is advantageous because it helps us work with an otherwise complicated differential equation, breaking it down into more manageable pieces by considering each term of the series independently.

Power Series Method

The power series method is a sophisticated technique for solving linear differential equations. Contrary to methods that may attempt to find solutions in terms of elementary functions, the power series method seeks solutions as an infinite sum of terms, each being a constant multiplied by a power of the variable.

In our step-by-step solution, the power series method begins by assuming that the solution y(x) is of the form \( y(x) = \sum_{n=0}^{\infty} a_n x^n \). This assumption is not without merit; power series are versatile since many functions can be represented as such, including complicated ones that are not easily expressible otherwise. After substitution of the assumed solution and its derivatives into the differential equation, coefficients of like powers of x are matched, leading to a condition each term in the series must satisfy, paving the path toward the recurrence relation.

Recurrence Relation

A recurrence relation is an equation that expresses the coefficients of a series in terms of previous coefficients. It plays an essential role in determining the coefficients of the power series solution for a differential equation. Once the power series and its derivatives are substituted back into the equation and the coefficients are matched, a pattern (or relation) emerges between subsequent coefficients.

In the given solution process, the recurrence relation allows us to express the coefficient \( a_{n+2} \) in terms of \( a_{n+1} \) and \( a_n \). It gives us a systematic way to find all the terms of the power series. By iterating this relation, starting from the initial coefficients, we construct the power series that represents the solution to the differential equation. Once the relation is established and we know the initial conditions or the first few coefficients, we can compute as many terms as necessary to get a good understanding of the solution's behavior or to identify patterns that may even let us write down a closed-form expression for the general solution.