Question

Find power series$$ \left(x^{3}-1\right) y^{\prime \prime}+x^{2} y^{\prime}+x y=0 $$ solutions in powers of \(x\) of each of the differential equations in Exercises.

Short answer

The power series solution for the given differential equation \((x^3-1)y''+x^2y'+xy=0\) is given by: \[y(x) = a_0(1 - x^2) + a_1x^4\] where \(a_0\) and \(a_1\) are constants.

Step-by-step solution

Formulating the power series representation of y(x)

Assume that the solution y(x) can be represented by a power series of the form: \[y(x) = \sum_{n=0}^{\infty} a_n x^n\]

Differentiating the power series

Differentiate y(x) with respect to x to obtain y'(x) and y''(x): \[y'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1}\] \[y''(x) = \sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}\]

Substituting the power series representation and their derivatives into the given differential equation

Substitute y(x), y'(x) and y''(x) into the given differential equation: \[(x^3 - 1)\left(\sum_{n=2}^{\infty} n(n-1) a_n x^{n-2}\right) + x^2\left(\sum_{n=1}^{\infty} n a_n x^{n-1}\right) + x\left(\sum_{n=0}^{\infty} a_n x^n\right) = 0\]

Finding coefficients using method of undetermined coefficients

To find the coefficients, we will set the coefficients of the corresponding powers of x equal on both sides of the equation. For each power of x, the coefficients must add up to zero. This will give us a system of equations that we can solve to find the values of the coefficients a_n. The coefficients on both sides of the equation must be equal, so we have: \[\sum_{n=2}^{\infty} n(n-1)a_n x^{n-2} - \sum_{n=2}^{\infty}n(n-1)a_n x^{n+1} + \sum_{n=1}^{\infty}n a_n x^{n+1} + \sum_{n=0}^{\infty} a_n x^{n+2} = 0\] Comparing the coefficients for each power of x, and solving the resulting equations for the coefficients: \(a_0 + a_2 = 0 \Rightarrow a_2 = -a_0\) \(a_3 = 0\) \(a_4 = a_1\) \(a_5 = 0\) For higher powers, the coefficients must equal zero because all the coefficients of x^6 and above are not involved in the equation: \(a_n = 0\) for \(n \geq 6\) Now we have found the coefficients of the power series solution of the given differential equation: \[y(x) = a_0(1 - x^2) + a_1x^4\]

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) form a fundamental part of mathematical analysis and have numerous applications in various fields like physics, engineering, and biology.
At their core, ODEs involve functions of one independent variable and the derivatives of that function. In simpler terms, they describe how something changes over time or space.
The specific type of ordinary differential equation you might encounter depends on its order, linearity, and the presence of constant coefficients.

• Order: The highest derivative that appears in the equation.
• Linearity: Determines if the equation is linear or non-linear. A linear ODE does not multiply differentials together or have any nonlinear terms like squares.
• Constant Coefficients: Whether the coefficients are constants or functions of the independent variable.

Understanding these basics is essential for tackling more complex problems like those involving power series. The problem above is a second-order linear ODE without constant coefficients.

Differential Equation Solving

Solving differential equations involves finding a function (or set of functions) that satisfies the equation. For many differential equations, closed-form solutions may not exist, but various methods can approximate solutions that are practically useful.

Among these methods, the most popular ones include:

• Separation of Variables: Used when you can separate the variables on two sides of the equation.
• Integration Factors: Primarily used for first-order linear ODEs.
• Variation of Parameters: For non-homogeneous differential equations.
• Power Series Methods: Especially useful for solving linear differential equations when solutions in elementary functions are not easy to find.

In our case, since we're using a power series, we convert the differential equation into an algebraic problem, which we solve by comparing coefficients.

Series Representation

Series representation, in the context of solving differential equations, refers to expressing the solution as an infinite sum of terms, usually involving powers of the independent variable. This approach is especially advantageous when dealing with equations difficult to solve in closed form.

The procedure usually involves:

• Expressing the solution as a series, like \(y(x) = \sum_{n=0}^{\infty} a_n x^n\).
• Differentiating the series to obtain expressions for \(y'(x)\) and \(y''(x)\).
• Substituting these series into the differential equation.

This transformation turns the problem into finding the coefficients \(a_n\) of the series.
Comparing coefficients of like powers of \(x\) yields equations that you solve to find these coefficients, as shown in the original step-by-step solution.

Method of Undetermined Coefficients

The Method of Undetermined Coefficients is used to find particular solutions to linear differential equations with constant coefficients or equations involving a series when the solution form can be guessed or hypothesized. In these cases, the solution is assumed rather than derived directly at the start.

This method can be broken into a few systematic steps:

• Assume a solution form (power series, for example).
• Substitute this assumed solution back into the differential equation.
• Equate coefficients of the same powers of the variable from both sides of the equation, forming a system of algebraic equations.
• Solve this system to find the coefficients in your assumed solution.

In our exercise, this leads us to identify which coefficients of the series must be zero, leaving us with a viable series representation of the solution.