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A power series is an infinite sum used to represent functions using terms based on increasing powers of a variable, in this case, the variable is \(x\). The expression for the power series is given by:

• \( y(x) = \sum_{k=0}^{\infty} a_k x^k \)

This representation assumes we can solve differential equations by expressing the unknown function as a series of powers of \(x\), with the coefficients \(a_k\) to be determined.
The process starts by writing the solution \(y(x)\) as a power series.

• This requires finding its derivatives, which are represented similarly as infinite sums.
• This makes it possible to substitute them back into the differential equation.

For a successful power series solution, these coefficients must satisfy the expanded differential equation. By using the power series expansion, the entire equation can be expressed in terms of corresponding powers of \(x\), allowing for an elegant solution method for complex differential equations.

Differential equations describe relationships involving derivatives, which indicate rates of change. Solving a differential equation means finding a function that satisfies this relation. Consider the differential equation:

• \( \left(1-x^{2}\right) y^{\prime \prime}-5 x y^{\prime}-4 y=0 \)

To solve this equation, we assumed a power series solution for \(y(x)\) and calculated its derivatives. By substituting the series representations of \(y(x)\), \(y'(x)\), and \(y''(x)\) into the equation, we can manipulate and rearrange the terms, collecting like terms together.
The solution involves equating coefficients of like powers of \(x\) to zero, ensuring that this sum holds true for all values of \(x\). This results in a recurrence relation that can be solved to find the series coefficients, \(a_k\), hence constructing the solution. The solution must balance out the equation for all terms to achieve a general solution.

In calculus, series representations are powerful tools for solving equations, including differential equations. They provide a way to express complex functions as sums based on simpler components. Generally, a function \(y(x)\) can be represented as a series:

• \( y(x) = \sum_{k=0}^{\infty} a_k x^k \)

Using series in calculus, particularly with Taylor and power series, is beneficial because:

• They simplify the solving processes of intricate differential equations.
• Series allow functions to be analyzed term-by-term, offering clear insights and solutions.

A series representation approximates a function by summing up infinite terms where each is associated with a particular power of \(x\). The utility in differential calculus comes from simplifying complex functions into manageable components, easing both computation and conceptual understanding.