91Ó°ÊÓ

A linear differential equation is one in which the dependent variable and all its derivatives appear linearly in the equation; that is, no powers, products, or other nonlinear functions of the variable or its derivatives are present. In the context of our second-order equation, we are looking at a differential equation involving the function, its first derivative, and second derivative, all in a linear format.

When dealing with a second-order differential equation, it typically has the form:

• \[ a(x)y'' + b(x)y' + c(x)y = f(x) \]

Here, \( a(x), b(x), \) and \( c(x) \) are functions of \( x \) and \( f(x) \) is a given function. In our exercise, \( f(x) = 0 \), making it a homogeneous linear differential equation.

The objective is to find an equation that holds for all solutions of the given form, \( y = c_1 x + c_2 x^2 \). This means the task involved writing the equation without reference to any particular values of \( c_1 \) or \( c_2 \), by utilizing linear combinations of the function and its derivatives.

The elimination of constants is a fundamental process when deriving a differential equation from a given family of solutions. Constants such as \( c_1 \) and \( c_2 \) in our solution function normally represent arbitrary parameters. These constants need to be eliminated to form a differential equation that is universally valid for all possible values of \( c_1 \) and \( c_2 \).

To do this, we perform differentiation:

• First, by differentiating \( y = c_1 x + c_2 x^2 \) to obtain its first derivative: \( y' = c_1 + 2c_2 x \).
• Then differentiate again to find the second derivative: \( y'' = 2c_2 \).

From these, we solve for \( c_2 = \frac{1}{2}y'' \) and substitute it back into \( y' \) to express in terms of the derivatives only, removing \( c_2 \).

Next, solve the adjusted \( y' \) to find \( c_1 = y' - xy'' \). This successfully expresses both constants in terms of \( y \) and its derivatives, leading to their elimination.

The concept of a two-parameter family of solutions is important in understanding differential equations. When we refer to such solutions, it means that the differential equation initially has a general solution that involves two arbitrary constants. These constants, \( c_1 \) and \( c_2 \), provide flexibility, representing a wide family of curves described by the equation.

The form \( y = c_1 x + c_2 x^2 \) indicates a solution generated by any value of these constants. Different pairs \( (c_1, c_2) \) will result in different specific solutions, each still valid within the broader solution family. This is an inherent property of solutions to second-order differential equations.

By manipulating the differential equation through differentiation and substitution, the initial general expression involving \( c_1 \) and \( c_2 \) becomes an equation void of them, indicating a specific relationship among the derivatives of \( y \) that must hold for any allowed solution within this family.