91Ó°ÊÓ

The second derivative of a function provides insights into the concavity of the original function. Calculating the second derivative is crucial in solving differential equations, like the one in our exercise. In this case, we have two functions, and our goal is to find whether each one satisfies the given differential equation. Understanding the role of the second derivative helps predict the function's behavior. To calculate a second derivative, you must first determine the first derivative of the function. By differentiating the first derivative, you then obtain the second derivative. For example, in the function \( y_1 = x - \ln x \), the first derivative is \( y_1' = 1 - \frac{1}{x} \). By taking the derivative of \( y_1' \), we find the second derivative \( y_1'' = \frac{1}{x^2} \). With these derivatives in hand, we proceed to verify if the original function solves the differential equation given.

The substitution method is a useful technique for verifying solutions to differential equations. This approach involves taking the derivatives of the proposed solutions and inserting them back into the original differential equation to check if they satisfy it. In our exercise, the substitution method is used for both proposed functions. For the first function \( y_1 \), after calculating its derivatives, they are substituted back into the differential equation: \[ x^2 y_1'' + x y_1' - y_1 = \ln x \]. If the left-hand side simplifies to equal the right-hand side, \( \ln x \), as shown in the example, the solution is verified. This substitution process allows you to systematically verify whether each function is indeed a solution to the given differential equation.

Logarithmic functions, like \( \ln x \), frequently appear in real-world applications and differential equations. In these contexts, understanding the basic properties and derivatives of logarithmic functions is crucial. Logarithms transform multiplicative processes into additive ones, making them particularly useful in various types of differential equations. In the problem at hand, our differential equation includes \( \ln x \), and the functions we examine, \( y_1 = x - \ln x \) and \( y_2 = \frac{1}{x} - \ln x \), both involve \( \ln x \). Knowing that the derivative of \( \ln x \) is \( \frac{1}{x} \) aids in computing the derivatives of the given functions. Thus, understanding logarithmic functions not only helps in performing differentiation but is also essential in verifying solutions in the context of this problem.

Verifying solutions is a critical step in solving differential equations. This step ensures that the functions proposed actually satisfy the given equation under the specified conditions. In the exercise, after computing derivatives through differentiation, each function is substituted back into the original differential equation. The goal is to see that:

• The specified differential equation's left-hand side,
  when plugged with the function and its derivatives, should equal the right-hand side.
• For both functions in the exercise, when substituted correctly, the end result of simplification on the left is \( \ln x \),
  identically equal to the right-hand side.

Successfully verifying solutions is like putting the finishing touch on a mathematical puzzle
where each piece falls into place with certainty.