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The Wronskian is a determinant used to assess whether a set of solutions is linearly independent. It is particularly important in solving systems of linear differential equations. For two functions, the Wronskian is represented as:

• \(W(y_1, y_2) = \begin{vmatrix} y_1 & y_2\ y_1' & y_2' \end{vmatrix}\).

In simpler terms, it checks if one function is a scalar multiple of another. If the Wronskian is not zero, the functions are linearly independent. For instance, in our original task:- The Wronskian for \( \frac{1}{t} \) and \( y_2(t) \) is the determinant of the matrix consisting of these functions and their derivatives. This determinant helps us explore whether \( y_2(t) \) truly offers a new solution different from \( \frac{1}{t} \).

Abel's formula is a powerful tool in the realm of differential equations, simplifying the process of finding a Wronskian. Specifically, it aids in determining the constancy of the Wronskian for linear differential equations of the form \( t^2y'' + 3ty' + y = 0 \).Abel's formula states:

• \[W(y_1, y_2) = c \cdot e^{\int -p(t)/t \cdot dt}\]

Here, \( p(t) \) is obtained from the coefficient of \( y' \) in the differential equation. This expression simplifies this problem, showing that the Wronskian is either zero or a constant. Knowing that the Wronskian is constant, you can use it to find a function \( v(t) \) for the second solution's formula. With the integral equaling \(-3 \ln(t) \), it informs us about the nature of \( v(t) \), guiding us to achieve:

• \( v(t) = \frac{3}{2t^2} \), making sure \( y_2(t) = v(t) \times y_1(t) \).

Linearly independent solutions are crucial when solving differential equations. They ensure diverse and non-redundant solutions, representing different directions or dimensions of the solution space.A set of solutions \( \{ y_1, y_2, \ldots, y_n \} \) is linearly independent if no solution can be expressed as a linear combination of others. The implication is profound:

• If combined appropriately, they can form a general solution for the differential equation.

In our context, we start with a known solution \( y_1(t) = \frac{1}{t} \). To ensure \( y_2(t) \) is independent, it must offer something new—thus the check via the Wronskian.Using the derivations:

• \( y_2(t) = \frac{3}{2t^3} \), multiplying the function \( v(t) \) by \( y_1(t) \), verifying distinctness.

This approach guarantees we have an appropriately independent solution in different behavioral states.