Question

Find the power series in \(x\) for the general solution. $$ \left(1+x^{2}\right) y^{\prime \prime}-8 x y^{\prime}+20 y=0 $$

Short answer

Based on the solution provided, write a short answer describing the power series representation of the general solution for the given second-order differential equation: The general solution of the given second-order linear differential equation can be represented as a power series: $$ y(x) = c_0 \sum_{n=0}^{\infty} c_n x^n + c_1 \sum_{n=1}^{\infty} c_n x^n, $$ where \(c_n\) are the coefficients of the power series, calculated using the recurrence relation: $$ c_{n+2} = \frac{8n-20+n(n-1)c_n}{(n+1)(n+2)}c_n. $$ The power series representation of the general solution allows us to approximate the solution using a polynomial of desired degree.

Step-by-step solution

Identify the Differential Equation Type

We have a second-order linear differential equation with variable coefficients given by: $$ \left(1+x^{2}\right) y^{\prime \prime}-8 x y^{\prime}+20 y=0 $$

Assume a Power Series Solution

We aim to find a power series solution of the form: $$ y(x) = \sum_{n=0}^{\infty} c_n x^n $$ where \(c_n\) are the coefficients of the power series.

Calculate the First and Second Derivatives

We need the first and second derivatives of \(y(x)\) to substitute into the given differential equation. Here are those derivatives: $$ y'(x) = \sum_{n=1}^{\infty} n c_n x^{n-1} $$ $$ y''(x) = \sum_{n=2}^{\infty} n(n-1) c_n x^{n-2} $$

Replace \(y\), \(y'\), and \(y''\) in the Differential Equation

Substitute the power series and its derivatives into the given differential equation and simplify: $$ \left(1+x^{2}\right) \sum_{n=2}^{\infty} n(n-1) c_n x^{n-2} - 8x \sum_{n=1}^{\infty} n c_n x^{n-1} + 20 \sum_{n=0}^{\infty} c_n x^n = 0 $$

Match the Power Series

We will rearrange the terms in such a way that each power of x lines up: $$ \sum_{n=2}^{\infty} n(n-1) c_n x^{n-2} + \sum_{n=2}^{\infty} n(n-1) c_n x^{n} - 8 \sum_{n=1}^{\infty} n c_n x^{n} + 20 \sum_{n=0}^{\infty} c_n x^n = 0 $$ Now we can combine the terms and find a recurrence relation: $$ \sum_{n=0}^{\infty}[(n+2)(n+1)c_{n+2}+ n(n-1)c_n - 8nc_n + 20c_n] x^n = 0 $$

Form a Recurrence Relation

To make the equation true for all values of x, we require that the coefficient of each \(x^n\) term be equal to zero. Thus, we have the recurrence relation: $$ (n+2)(n+1)c_{n+2}+ n(n-1)c_n - 8nc_n + 20c_n = 0 $$ Simplify and rearrange for the coefficients: $$ c_{n+2} = \frac{8n-20+n(n-1)c_n}{(n+1)(n+2)}c_n $$

Calculate a Few Coefficients (Optional)

Using the determined recurrence relation, we can obtain some coefficients of the power series (this might not be necessary but could be useful for visualizing the first few terms of the power series): $$ c_2 = -\frac{20}{2}c_0 = -10c_0 $$ $$ c_3 = \frac{20}{6}c_1 = \frac{10}{3}c_1 $$ and so on.

Write the General Power Series Solution

Finally, we can write down the general power series solution as: $$ y(x) = c_0 \sum_{n=0}^{\infty} c_n x^n + c_1 \sum_{n=1}^{\infty} c_n x^n $$ This is the power series for the general solution of the given differential equation.

Key concepts

Understanding Second-Order Linear Differential Equations

When we talk about second-order linear differential equations, we're discussing a cornerstone of differential equations applied in physics, engineering, and applied mathematics. These equations are characterized by the highest derivative being second-order and the equation being linear in the dependent variable and its derivatives. Linear implies that the equation can be written in the form
\[\begin{equation}a_2(x)y'' + a_1(x)y' + a_0(x)y = g(x),\br\end{equation}\]
where the function g(x) is known and the coefficients \(a_2(x)\), \(a_1(x)\), and \(a_0(x)\) are functions of the independent variable x. In the context of our exercise, \(a_2(x)=1+x^2\), \(a_1(x)=-8x\), and \(a_0(x)=20\). The equation does not contain any products or powers of y and its derivatives, keeping the terms linear. Power series solutions, like the one we're seeking, come in handy especially when the coefficients are not constant, and standard methods (such as the characteristic equation) don't apply.

Navigating Through Recurrence Relations

A recurrence relation is a mathematical formulation that expresses each term of a sequence as a function of its preceding terms. In the context of solving differential equations using power series, it allows us to systematically determine the coefficients \(c_n\) of the series. The relation arises naturally when we equate coefficients of like powers of x after substituting the power series (and its derivatives) into the differential equation.

• We aim to set terms multiplying each power of x to zero because we want the equation to hold for all x – a requirement for our power series to be a solution.
• The recurrence relation we find gives us a recipe: how to build more complex series terms from simpler ones such as \(c_{n+2}\) being computed using \(c_n\).

Our exercise gives rise to the recurrence relation
\[\begin{equation}c_{n+2} = \frac{8n - 20 + n(n-1)c_n}{(n+1)(n+2)}c_n,\br\end{equation}\]
which lets us determine all the coefficients of the power series once we have a couple of initial conditions, \(c_0\) and \(c_1\) usually found from initial or boundary value conditions.

Coefficients of Power Series and Solution Patterns

The coefficients of a power series are the constants \(c_n\) that multiply each term of the form \(x^n\). They determine the behavior and convergence properties of the series. In solving differential equations, once we’ve established a recurrence relation, it becomes possible to compute these coefficients methodically.

• Two initial coefficients are typically required to generate the full power series using the recurrence relation.
• These coefficients can often be tied to initial/boundary conditions for the specific physical situation modeled by the differential equation.
• By calculating them, we capture the essence and detail of the solution to the differential equation.

Visualizing Patterns

It is not just about finding a sequence of numbers, but about recognizing patterns that allow us to express complex phenomena in terms of an infinite sum, better known as a power series. Understanding the role of these coefficients within the power series grants us not only the solution but also significant insight into the behavior of solutions to differential equations concurrently. In practical terms, by resolving for a few coefficients as suggested in Steps 6 and 7 of our solution, such as \(c_2\) and \(c_3\), we can start discerning this pattern and by extension, predict the form of subsequent coefficients.