91Ó°ÊÓ

Differential equations are mathematical statements that relate a function with its derivatives. They are often used to describe various dynamically changing systems. In essence, a differential equation will involve one or more unknown functions and their derivatives.
These equations come in many forms: first-order, second-order, linear, non-linear, ordinary, and partial, among others. The order of a differential equation is determined by the highest derivative present. For instance, the equation given in the exercise is a second-order linear differential equation because the highest derivative is the second one.
Solving differential equations often involves determining a function that satisfies the equation under given conditions. This process sometimes requires finding the general solution, which incorporates all possible solutions of the differential equation.

A homogeneous differential equation is one where the function and all its derivatives equate to zero, as seen in the homogeneous form of the given equation: \( x^2 y'' - 2xy' + 2y = 0 \).
In this type of equation, substitution is often needed to solve for the solutions. Homogeneous equations are critical because they allow us to find the complementary solution, denoted as \( y_c \). This particular solution involves no external forcing function.
The exercise reveals a homogeneous equation in the step involving the Euler-Cauchy method, which typically involves solving based on characteristic equations. The resulting roots, in this case being \( r_1 = 1 \) and \( r_2 = 2 \), help in forming the general solution, \( y_c = C_1 x + C_2 x^2 \). These solutions form a foundation upon which the particular solution is built.

The Wronskian is a determinant useful for determining the linear independence of a set of solutions. In this exercise, it's calculated to ensure that the functions \( y_1 = x \) and \( y_2 = x^2 \) are linearly independent.
The Wronskian is crucial when we employ the method of variation of parameters. Here, the Wronskian \( W(y_1, y_2) \) of\( y_1 = x \) and \( y_2 = x^2 \) is calculated as \[W(y_1, y_2) = \begin{vmatrix} y_1 & y_2 \ y'_1 & y'_2 \end{vmatrix}\]which simplifies to \[x \cdot 2x - x^2 \cdot 1 = x^3\]The Wronskian, being nonzero (\( x^3 \)), confirms that \( y_1 \) and \( y_2 \) are functionally unique and can be used in constructing solutions within the variation of parameters.

Integration by parts is a fundamental technique in calculus used to solve integrals where direct integration is complex or not possible. This technique is based on the formula: \\[ \int u \, dv = uv - \int v \, du \]
This concept was used in the exercise to determine \( u_1(x) \) and \( u_2(x) \) in the step-by-step solution. When solving \( \int x^2 \ln x \, dx \) and \( \int x \ln x \, dx \), integration by parts helps break down the complex integral into simpler parts.
For example, if you choose \( u = \ln x \) and \( dv = x^2 \, dx \), then \( du = \frac{1}{x} \, dx \) and \( v = \frac{x^3}{3} \). This substitution helps in evaluating the integral piece by piece.
Similar steps are followed to find \( \int x \ln x \, dx \), simplifying the integration process and yielding the required functions needed for solving the differential equation thoroughly.