Question

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

Short answer

Question: Determine the power series representation of the general solution of the given differential equation: \(\left(1+x^{2}\right) y^{\prime \prime}+2 x y^{\prime}-2 y=0\). Answer: The general power series solution is given by: $$ y(x) = a_0\left(\sum_{n=0}^{\infty} a_nx^n \right)+ a_1\left(\sum_{n=1}^{\infty} na_nx^{n-1}\right), $$ where \(a_0\) and \(a_1\) are constants determined by the initial conditions of the problem.

Step-by-step solution

Write the power series representation of y(x)

Write the general solution \(y(x)\) as a power series in \(x\): $$ y(x) = \sum_{n=0}^{\infty} a_nx^n $$ where \(a_n\) are the coefficients of the power series.

Compute the derivatives of y(x)

Find the first and second derivatives, \(y'(x)\) and \(y''(x)\), by differentiating the power series term by term: $$ y'(x) = \sum_{n=1}^{\infty} na_nx^{n-1} $$ $$ y''(x) = \sum_{n=2}^{\infty} n(n-1)a_nx^{n-2} $$

Substitute the power series of y(x), y'(x), and y''(x) into the given differential equation

Substitute the power series expressions for \(y(x)\), \(y'(x)\), and \(y''(x)\) into the given differential equation: $$ \left(1+x^{2}\right) \left(\sum_{n=2}^{\infty} n(n-1)a_nx^{n-2}\right) + 2x\left(\sum_{n=1}^{\infty} na_nx^{n-1}\right) - 2\left(\sum_{n=0}^{\infty} a_nx^n\right) = 0 $$

Equate coefficients of like terms

Multiply out and equate coefficients of like terms in the power series: $$ \sum_{n=2}^{\infty} n(n-1)a_nx^{n-2} + \sum_{n=0}^{\infty} n(n-1)a_nx^{n} + 2\sum_{n=1}^{\infty} na_nx^n - 2\sum_{n=0}^{\infty} a_nx^n = 0 $$ Since the power series representation must hold true for all \(x\), the coefficients in the series must be zero: $$ n(n-1)a_n + n(n-1)a_n + 2na_n - 2a_n = 0 \quad \text{for all } n \geq 0 $$

Determine the recursion relation for the coefficients

Rearrange the above equation to solve for the coefficients \(a_n\): $$ a_n \left[2n(n-1)+2n-2\right] = 0 $$ \(a_n\) cannot be zero for all \(n \geq 0\), otherwise, \(y(x) = 0\) would not be a nontrivial solution. Therefore, the expression inside the brackets must be equal to zero: $$ 2n(n-1)+2n-2 = 0 \Rightarrow n = 1 $$ Since the recursion relation depends on \(n\), it is the initial conditions \(a_0\) and \(a_1\) that will determine the unique solution for this problem. Thus, the general power series solution is given by: $$ y(x) = a_0\left(\sum_{n=0}^{\infty} a_nx^n \right)+ a_1\left(\sum_{n=1}^{\infty} na_nx^{n-1}\right) $$

Key concepts

Power Series

Power series are infinite sums of terms in the form of variables raised to successive powers and multiplied by coefficients. They provide a robust way to represent functions and, when applied to differential equations, can help find solutions where other techniques might not work.
Imagine a power series as an extended polynomial that includes infinitely many terms:

• The general form is \(y(x) = \sum_{n=0}^{\infty} a_nx^n\), where \(a_n\) represents the sequence of coefficients, and \(x^n\) denotes the variable \(x\) raised to the power of \(n\).
• Power series solutions are particularly useful in handling linear differential equations, especially near ordinary points, where functions can be tough to solve via traditional methods.

Using power series, we can transition from complex differential equations to simpler polynomial equations by expressing solutions as sums of infinite series.

General Solution

In differential equations, finding a general solution means deriving a formula that encapsulates all possible solutions of the equation. For a given differential equation, the general solution is usually achieved by determining and combining all the possible power series solutions.

• A differential equation might be represented in terms of a power series, allowing us to express the solution as infinite sums.
• The general solution to the equation preserves all variables and parameters, which can later be specified or adjusted based on boundary or initial conditions.

In our exercise, the power series solution connects the derivatives of the function with infinite sums, ultimately forming the general solution of the differential equation.

Recursion Relation

Recursion relation in differential equations refers to an equation that connects terms in a sequence or series to its predecessors. It is a systematic way to establish each term based on the previous one, a crucial step in solving equations via power series.
For instance:

• In our problem, after substituting the power series into the differential equation and performing algebraic manipulation, a recursion relation emerges connecting the coefficients.
• Specifically, the relation \(2n(n-1)+2n-2 = 0\) identifies the pattern among power series coefficients.

By solving this recursion relation, we identify how each term of the series depends on its previous ones, helping to construct the full power series solution.

Coefficients

The coefficients in a power series are crucial as they dictate the impact of each \(x^n\) term in the expansion. They play an essential role in shaping the function defined by the series and solving differential equations.

• In our solution process, the coefficients \(a_n\) are determined by equating like terms and solving resulting equations based on the derived recursion relation.
• The initial coefficients \(a_0\) and \(a_1\) are specifically crucial because they serve as initial conditions, influencing the series from which other coefficients derive.

Thus, solving for coefficients involves careful attention to how they contribute to each power's term, ensuring the function represented by the series satisfies the differential equation.