91Ó°ÊÓ

A linear second-order differential equation is a type of differential equation characterized by a second derivative and linearity in terms of the dependent variable and its derivatives. The equation generally takes the form:

• \( a(x) \frac{d^2y}{dx^2} + b(x) \frac{dy}{dx} + c(x)y = g(x) \)

where \( a(x) \), \( b(x) \), and \( c(x) \) are functions of the independent variable \( x \), and \( g(x) \) is a function that may be zero, indicating a homogeneous equation.
In the exercise, the given equation \( y'' - 2(x+3)y' - 3y = 0 \) is homogeneous as \( g(x) = 0 \). Solving these equations often involves transformation techniques, especially when coefficients depend on the independent variable, making them more complex to tackle directly.

To solve a differential equation like the one in the exercise, we can use a series solution. A series solution involves expressing the solution as an infinite sum of terms, typically a power series.
This approach is particularly useful when the differential equation has variable coefficients and is centered around a point of interest, which in our case, is \( x = -3 \). The goal is to write the solution in the form:

• \( y(z) = \sum_{n=0}^{\infty} a_n z^n \)

where \( a_n \) are constants to be determined. The series solution method leverages calculus, specifically the ability to differentiate and integrate power series term-by-term, providing a powerful tool to obtain approximations of functions that solve the differential equation in a local interval around the point of interest.

The change of variable is a strategic tool used to simplify the form of a differential equation, making it easier to solve. By changing the variable, we center our focus on a specific point of interest, which can simplify the coefficients of the differential equation.
In the given problem, a change of variables is applied by letting \( z = x + 3 \). This shifts the equation to be centered around \( z = 0 \). Consequently, the differential equation \( y'' - 2(x+3)y' - 3y = 0 \) transforms to:

• \( \frac{d^2 y}{dz^2} - 2z \frac{dy}{dz} - 3y = 0 \)

This transformation simplifies calculations and allows us to pursue a series solution more easily by focusing around the newly defined center \( z = 0 \). By doing so, we align the problem to suit power series expansion, effectively managing the complexities introduced by variable coefficients.

A power series is an infinite series of the form \( \sum_{n=0}^\infty a_n z^n \), where \( a_n \) are coefficients and \( z^n \) term contributes increasingly higher powers of \( z \). This kind of series is a flexible mathematical tool used to represent functions analytically, especially near a point where the function behaves in an analytic manner.
In the context of solving differential equations, power series allow us to express solutions in terms of simple, polynomial-like components. We assume a solution of the form:

• \( y(z) = \sum_{n=0}^\infty a_n z^n \)

We then derive recurrence relations by substituting this series along with its derivatives back into the differential equation, solving for the coefficients \( a_n \). This process captures the behavior of solutions locally, producing a valid approximate solution near the specified point (for instance \( z = 0 \) in our exercise). This approach is essential when exact solutions are either difficult or impossible to find directly.