Final answer:
To verify that the given functions form a fundamental set of solutions of the differential equation, we need to show that they satisfy the differential equation and are linearly independent.
Step-by-step explanation:
To verify that the given functions form a fundamental set of solutions of the differential equation, we need to show that they satisfy the differential equation and are linearly independent. Let's start by substituting the functions into the differential equation:
x²y'' + xy' + y = 0
Substituting cos(ln(x)):
x²(-sin(ln(x)) + cos(ln(x))) + x(cos(ln(x))) + cos(ln(x)) = 0
x²(-sin(ln(x))) + x(cos(ln(x))) + x(cos(ln(x))) + cos(ln(x)) = 0
-x²sin(ln(x)) + 2x(cos(ln(x))) + cos(ln(x)) = 0
This equation holds true for all values of x, so cos(ln(x)) is a solution to the differential equation. Now let's substitute sin(ln(x)):
x²(-cos(ln(x)) + sin(ln(x))) + x(sin(ln(x))) + sin(ln(x)) = 0
x²(-cos(ln(x))) + x(sin(ln(x))) + x(sin(ln(x))) + sin(ln(x)) = 0
-x²cos(ln(x)) + 2x(sin(ln(x))) + sin(ln(x)) = 0
This equation holds true for all values of x, so sin(ln(x)) is also a solution to the differential equation.
To show that the functions are linearly independent, we can calculate their Wronskian:
W(cos(ln(x)),sin(ln(x))) = cos(ln(x))(sin(ln(x)))' - sin(ln(x))(cos(ln(x)))' = cos(ln(x))(cos(ln(x)) / x) - sin(ln(x))(-sin(ln(x)) / x) = cos²(ln(x)) / x + sin²(ln(x)) / x = 1 / x
The Wronskian is non-zero for x > 0, which means the functions are linearly independent. Therefore, the given functions form a fundamental set of solutions of the differential equation on the indicated interval.