Question

Determine whether the given functions form a fundamental set of solutions for the linear system. $$ \text { 4. } \mathbf{y}^{\prime}=\left[\begin{array}{cc} 0 & 1 \\ t^{-2} & -t^{-1} \end{array}\right] \mathbf{y}, \quad \mathbf{y}_{1}(t)=\left[\begin{array}{l} t \\ 1 \end{array}\right], \quad \mathbf{y}_{2}(t)=\left[\begin{array}{c} t^{-1} \\ -t^{-2} \end{array}\right], \quad 0

Short answer

Short Answer: Yes, the given solutions form a fundamental set of solutions for the linear system, as the calculated Wronskian is nonzero for the given range of \(t\).

Step-by-step solution

Calculate the Wronskian

First, we will calculate the Wronskian (W) of the given solutions, \(\mathbf{y}_1\) and \(\mathbf{y}_2\). The Wronskian for a 2x2 system is given by the determinant of the matrix formed by these solutions: \[ W(\mathbf{y}_1,\mathbf{y}_2)=\left|\begin{array}{cc}y_{1,1} & y_{2,1}\\y_{1,2} & y_{2,2}\end{array}\right| \] In our case, the Wronskian is: \[ W(\mathbf{y}_1,\mathbf{y}_2)=\left|\begin{array}{cc}t & t^{-1}\\1 & -t^{-2}\end{array}\right| \]

Evaluate the determinant of the Wronskian matrix

Now, we will evaluate the determinant of the Wronskian matrix: \[ W(\mathbf{y}_1,\mathbf{y}_2)=t(-t^{-2})-(t^{-1})(1) = -t^{-1}-t^{-1} = -2t^{-1} \]

Check if the Wronskian is nonzero

We have found the Wronskian to be -2t^{-1}. Now, we need to check whether this function is nonzero for the given range of \(t\), which is 0<t<\(\infty\). Since \(t\) is always positive for the given range, it's clear that the Wronskian will also be nonzero for \(0<t<\infty\).

Conclusion

Since the Wronskian of the given solutions is nonzero for the given range of \(t\), the functions \(\mathbf{y}_1(t)\) and \(\mathbf{y}_2(t)\) form a fundamental set of solutions for the linear system.

Key concepts

Wronskian

The Wronskian is a key tool in the study of differential equations and provides information about the linear independence of a set of functions. When dealing with sets of solutions to a differential equation, the Wronskian can be utilized to determine whether the functions comprise a fundamental set of solutions. Formally, the Wronskian of two functions, let's say, \(f\) and \(g\), is defined as the determinant of the matrix that contains the functions and their derivatives:
\[ W(f,g)=\begin{vmatrix} f & g \ f' & g' \end{vmatrix} \].
For functions that are solutions to a linear system of differential equations, if the Wronskian is nonzero at some point within the interval of interest, it implies that the set of solutions is linearly independent. This linear independence in turn establishes that we have a fundamental set of solutions to the differential equation on that interval, laying the foundation for expressing the general solution as a linear combination of these fundamental solutions.

Linear Systems of Differential Equations

Linear systems of differential equations consist of multiple differential equations involving the same set of dependent variables. A system is said to be 'linear' if the dependent variables and their derivatives appear to the power of one and are not multiplied or composed with each other.

For instance, a two-dimensional linear system can be represented as:
\[ \begin{aligned}\frac{dy_1}{dt} &= a_{11}(t)y_1 + a_{12}(t)y_2, \frac{dy_2}{dt} &= a_{21}(t)y_1 + a_{22}(t)y_2, \end{aligned}\]
where \(a_{ij}(t)\) are functions of \(t\) but not of \(y_1\) or \(y_2\). Solutions to these systems are sought in the form of vector functions, and these solutions can be understood better through the concept of the fundamental set. If we can find a sufficient number of independent solutions to span the solution space, we can construct any solution to the system from these 'fundamental' solutions.

Determinant of a Matrix

The determinant of a matrix is a special scalar value that provides important information about the matrix and the linear transformations it represents. Typically denoted as \(\det(A)\) or \(|A|\), the determinant can tell us, for example, if a matrix is invertible or what the volume scaling factor of a linear transformation is.

In the context of the Wronskian and differential equations, the determinant is used to assess whether sets of solutions are linearly independent. For a 2x2 matrix \(A\), the determinant is found by:
\[ \det(A) = a_{11}a_{22} - a_{12}a_{21} \].
This corresponds to the computation we do when finding the Wronskian of two solution functions to a differential equation. If the determinant of their associated matrix (the Wronskian matrix) is non-zero, then the solutions are linearly independent and form a fundamental set for the system's general solution.

Homogeneous Equations

Homogeneous equations play a central role in the theory of linear differential equations and their systems. An equation is termed 'homogeneous' when all terms are a product of the dependent variable and its derivatives, and the equation equals zero.

For example, a homogenous linear differential equation of the second order would look like:
\[ ay'' + by' + cy = 0 \],
where \(a\), \(b\), and \(c\) are coefficients that may depend on the independent variable but not on \(y\) or its derivatives. In a system, the homogeneity implies that one can scale solutions and still obtain another solution, which is why the concept pairs well with the principles of linear superposition and independence. Furthermore, the solutions to homogeneous systems are particularly significant as they form the complementary part of the general solution, to which particular solutions of any non-homogeneous part can be added.