91Ó°ÊÓ

In differential equations, a change of variable is a technique used to simplify the equation to make it solvable. This technique involves substituting variables to transform the equation into a simpler or more recognizable form.
In the given problem, the original variable is transformed from \(x\) to \(z\) by implementing the change of variable \(z = x - 3\). This transformation simplifies the manipulations required to solve the equation.
By changing variables, the equation changes form, usually allowing easier identification and solving, especially with known methods such as the Cauchy-Euler approach.

• For instance, derivatives also change under this transformation: \( y'(x) \) becomes \( y'(z) \) and \( y''(x) \) becomes \( y''(z) \).

This simplifies the structure of the problem, making it easier to apply known solution techniques.

The standard form of a differential equation is a predefined way to equally represent linear homogeneous differential equations. The general form is \(a(x)y''(x) + b(x)y'(x) + c(x)y(x) = 0\).
This form is significant because it sets the stage for various methods to solve differential equations, such as series or transformation techniques, by providing a consistent structure.
In our exercise, the equation is already in standard form, which simplifies the task. This means that each term involving the dependent variable and its derivatives can be identified immediately:

• \(a(x) = 2(x-3)^2\),
• \(b(x) = 5(x-3)\),
• \(c(x) = -2\).

Recognizing the equation in this form allows us to apply the Cauchy-Euler method directly. This further strengthens our ability to determine a viable solution quickly.

The general solution of a differential equation is crucial as it provides the set of all possible solutions. It usually involves constant parameters that can be adjusted to account for initial conditions or specific requirements.
For a Cauchy-Euler equation like \(2z^2 y''(z) + 5z y'(z) - 2y(z) = 0\), the approach is to propose a solution of the form \(y(z) = z^m\).
When this form is substituted into the differential equation, it typically results in a quadratic characteristic equation for \(m\). Solving this equation provides the values of \(m\), which are essential in constructing the general solution.

• The solution becomes a linear combination of functions based on the found \(m\) values. For instance, it might look like \(y(z) = C_1z^{m_1} + C_2z^{m_2}\), where \(C_1\) and \(C_2\) are arbitrary constants.

Finally, substituting back the original variable, here \(z = x - 3\), gives the general solution in terms of \(x\), completing the process for this specific differential equation.