Question

(a) Verify that the given functions are solutions of the homogeneous linear system. (b) Compute the Wronskian of the solution set. On the basis of this calculation, can you assert that the set of solutions forms a fundamental set? (c) If the given solutions form a fundamental set, state the general solution of the linear homogeneous system. Express the general solution as the product \(\mathbf{y}(t)=\) \(\Psi(t) \mathbf{c}\), where \(\Psi(t)\) is a square matrix whose columns are the solutions forming the fundamental set and \(\mathbf{c}\) is a column vector of arbitrary constants. (d) If the solutions form a fundamental set, impose the given initial condition and find the unique solution of the initial value problem. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rrr} -2 & 0 & 0 \\ 0 & 1 & 4 \\ 0 & -1 & 1 \end{array}\right] \mathbf{y}, \quad \mathbf{y}(0)=\left[\begin{array}{r} 3 \\ 4 \\ -2 \end{array}\right] $$

Short answer

Question: Verify if the given functions are solutions to the given linear homogeneous system of ODEs, and if they form a fundamental set, determine the unique solution to the corresponding initial value problem. Functions: \(\mathbf{y}_1(t)\), \(\mathbf{y}_2(t)\), and \(\mathbf{y}_3(t)\) System of ODEs: \(-2\mathbf{y}_1(t) = \frac{\mathrm{d}\mathbf{y}_1(t)}{\mathrm{d}t}\), \( \mathbf{y}_2(t) + 4 \mathbf{y}_3(t) = \frac{\mathrm{d}\mathbf{y}_2(t)}{\mathrm{d}t}\), and \( -\mathbf{y}_2(t) + \mathbf{y}_3(t) = \frac{\mathrm{d}\mathbf{y}_3(t)}{\mathrm{d}t}\). Initial condition: \(\mathbf{y}(0) = \left[\begin{array}{r}3\\4\\-2\end{array}\right]\).

Step-by-step solution

Verify the given functions as solutions

To check if the given functions are solutions of the homogeneous linear system, we need to compute their derivatives and check if they satisfy the given system of ODEs. Let's call the given functions as \(\mathbf{y}_1(t), \mathbf{y}_2(t), \mathbf{y}_3(t)\). Compute the derivatives of \(\mathbf{y}_1(t)\), \(\mathbf{y}_2(t)\), and \(\mathbf{y}_3(t)\): \( \frac{\mathrm{d}\mathbf{y}_1(t)}{\mathrm{d}t}\), \(\frac{\mathrm{d}\mathbf{y}_2(t)}{\mathrm{d}t}\), and \( \frac{\mathrm{d}\mathbf{y}_3(t)}{\mathrm{d}t}\). Check if they satisfy the system of ODEs: \( -2\mathbf{y}_1(t) = \frac{\mathrm{d}\mathbf{y}_1(t)}{\mathrm{d}t}\), \( \mathbf{y}_2(t) + 4 \mathbf{y}_3(t) = \frac{\mathrm{d}\mathbf{y}_2(t)}{\mathrm{d}t}\), and \( -\mathbf{y}_2(t) + \mathbf{y}_3(t) = \frac{\mathrm{d}\mathbf{y}_3(t)}{\mathrm{d}t}\). If all three equations are true for the given functions, they are solutions to the system.

Compute the Wronskian and check for fundamental set

Compute the Wronskian of the solution set, which is the determinant of the matrix formed by the solutions: \(W(t) = \det(\Psi(t))\), where \(\Psi(t)\) is a matrix with columns \(\mathbf{y}_1(t), \mathbf{y}_2(t), \mathbf{y}_3(t)\). If \(W(t) \neq 0\) for all values of \(t\), then the given solutions form a fundamental set.

Find the general solution

If the given solutions form a fundamental set, the general solution of the linear homogeneous system can be written as: \(\mathbf{y}(t) = \Psi(t) \mathbf{c}\), where \(\mathbf{c}\) is a column vector of arbitrary constants. Here, \(\Psi(t)\) is the square matrix whose columns are the fundamental set, i.e., \(\mathbf{y}_1(t), \mathbf{y}_2(t), \mathbf{y}_3(t)\).

Apply the initial condition and find the unique solution

If the solutions form a fundamental set, we can use the given initial condition to find the unique solution of the initial value problem. Given: \(\mathbf{y}(0) = \left[\begin{array}{r}3\\4\\-2\end{array}\right]\). Use this to solve for the constants in \(\mathbf{c}\): \(\mathbf{y}(0) = \Psi(0) \mathbf{c}\). Once we find the constants in \(\mathbf{c}\), we can plug them back into the general solution \(\mathbf{y}(t) = \Psi(t) \mathbf{c}\) to get the unique solution of the initial value problem.

Key concepts

Homogeneous Linear Systems

Homogeneous linear systems are a fundamental concept in differential equations. These systems are a set of linear equations where every equation is set to zero. In mathematical terms, a homogeneous system can be expressed in matrix form as \( A \mathbf{y} = \mathbf{0} \). Here, \( A \) is a matrix of coefficients, \( \mathbf{y} \) is the vector of unknown functions, and \( \mathbf{0} \) is the zero vector.

For our specific exercise, the system of equations is

• \( -2 \mathbf{y}_1(t) = \frac{\mathrm{d}\mathbf{y}_1(t)}{\mathrm{d}t} \)
• \( \mathbf{y}_2(t) + 4 \mathbf{y}_3(t) = \frac{\mathrm{d}\mathbf{y}_2(t)}{\mathrm{d}t} \)
• \( -\mathbf{y}_2(t) + \mathbf{y}_3(t) = \frac{\mathrm{d}\mathbf{y}_3(t)}{\mathrm{d}t} \)

The goal is to determine if the given functions are solutions to this system. By verifying if the derivatives of these functions satisfy each equation, we confirm their validity as solutions. A key characteristic of homogeneous systems is that any linear combination of solutions is also a solution.

Wronskian

The Wronskian is a determinant used to determine if a set of solutions forms a fundamental set. It is an essential concept in analyzing linear differential equations. For a system of functions \( \mathbf{y}_1(t), \mathbf{y}_2(t), \ldots, \mathbf{y}_n(t) \), the Wronskian is represented as \( W(t) = \det(\Psi(t)) \), where \( \Psi(t) \) is a matrix whose columns are the functions \( \mathbf{y}_i(t) \).

In our problem, computing the Wronskian involves evaluating the determinant of the matrix formed by the given solutions. A non-zero Wronskian, \( W(t) eq 0\), for all \( t \), signifies that the solutions are linearly independent and therefore, form a fundamental set.

This determinative test is crucial as it reveals the capability of the solution set to span the solution space of the differential equation system, ensuring completeness and uniqueness in solutions.

Initial Value Problem

An initial value problem involves finding a specific solution to a differential equation that meets given conditions at a particular point. Typically, this involves solving an equation \( \mathbf{y}' = A \mathbf{y} \) with an initial condition \( \mathbf{y}(0) = \mathbf{y}_0 \).

In the exercise, we are asked to find the unique solution to the initial value problem given \( \mathbf{y}(0) = \begin{bmatrix} 3 \ 4 \ -2 \end{bmatrix} \). This means that the general solution \( \mathbf{y}(t) = \Psi(t) \mathbf{c} \) must be adjusted to satisfy this initial condition.

Once the initial condition is applied, it provides a way to solve for the constants in the vector \( \mathbf{c} \). Substituting these constants back into the general solution gives the specific solution that satisfies the system at the initial point.

Fundamental Set of Solutions

A fundamental set of solutions is a group of solutions to a differential equation that is linearly independent and spans the solution space of the equation. In practical terms, if you have a fundamental set, you can express the general solution of the system as a linear combination of the solutions in this set.

For our linear homogeneous system, a fundamental set is determined using the Wronskian. If the Wronskian is non-zero, we confirm that the given solutions, \( \mathbf{y}_1(t), \mathbf{y}_2(t), \mathbf{y}_3(t) \), form such a set.

This allows the general solution to be expressed as \( \mathbf{y}(t) = \Psi(t) \mathbf{c} \), where \( \Psi(t) \) is the matrix consisting of the fundamental solutions as columns, and \( \mathbf{c} \) is a vector of arbitrary constants. This method ensures a comprehensive understanding of the solutions' behavior and provides a complete solution framework for the initial value problem.