Question

In Exercises \(11-20\) find a particular solution, given that \(Y\) is a fundamental matrix for the complementary system. $$ \mathbf{y}^{\prime}=\frac{1}{2 t^{4}}\left[\begin{array}{cc} 3 t^{3} & t^{6} \\ 1 & -3 t^{3} \end{array}\right] \mathbf{y}+\frac{1}{t}\left[\begin{array}{c} t^{2} \\ 1 \end{array}\right] ; \quad Y=\frac{1}{t^{2}}\left[\begin{array}{cc} t^{3} & t^{4} \\ -1 & t \end{array}\right] $$

Short answer

Question: Given a system of first-order linear differential equations in matrix form with a fundamental matrix Y, find a particular solution using the provided Y. Answer: Unfortunately, the determinant of the provided fundamental matrix Y is zero, which implies that Y is not invertible. Fundamental matrices should always have a nonzero determinant. Therefore, there must be an error in the given Y. Without a correct fundamental matrix, we cannot proceed to find a particular solution to the system.

Step-by-step solution

1. Find the inverse of the given fundamental matrix Y.

To find the inverse of the given fundamental matrix, we will use the formula for the inverse of a 2x2 matrix: $$Y^{-1}(t) = \frac{1}{\det(Y(t))} \left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right]$$ where \(a, b, c, d\) are the elements of the original matrix. In our example: $$Y(t) = \frac{1}{t^2}\left[\begin{array}{cc} t^3 & t^4 \\ -1 & t \end{array}\right]$$ \(a=t^3, b=t^4, c=-1, d=t\). Now, we find the determinant and the inverse matrix: $$\det(Y(t)) = (t^3)(t) - (-1)(t^4) = t^4 - t^4 = 0$$ The determinant is zero, which means that the given fundamental matrix Y is not invertible. This cannot be correct as the fundamental matrix should always have a nonzero determinant. Then, it seems there is an error in the given Y. Without a correct fundamental matrix, we cannot continue with this problem.

Key concepts

Particular Solution

A particular solution to a differential equation is a solution that satisfies the non-homogeneous part of the equation. Unlike the general solution, which encompasses all possible solutions, a particular solution represents a specific case. In the context of systems of differential equations, finding a particular solution involves identifying a function that fits the system's inhomogeneous components.
Particularly in differential equations, we often utilize techniques like the variation of parameters or the method of undetermined coefficients to find these solutions. The particular solution will usually supplement the complementary (or homogeneous) solution to form the general solution.
In practice, this means that after finding the homogeneous solution, represented by the complementary system, the next step is to determine a function that, when combined with this solution, satisfies the entire differential equation.

Fundamental Matrix

A fundamental matrix, in the context of systems of linear differential equations, is a square matrix whose columns are linearly independent solutions to the system. It plays a crucial role in understanding the system's behavior and deriving other types of solutions.
The fundamental matrix can be used to express the general solution of a differential system. When you know a fundamental matrix, you can easily construct all solutions to the system's homogeneous equations. This is done by taking linear combinations of the columns of the fundamental matrix.
In mathematical terms, if \( Y(t) \) is a fundamental matrix for the differential system \( \mathbf{y}' = A(t) \mathbf{y} \), then the general solution is \( \mathbf{y}(t) = Y(t) \mathbf{c} \) for some constant vector \( \mathbf{c} \).
It's important to note that a fundamental matrix is not unique; any matrix obtained by multiplying a fundamental matrix by an invertible constant matrix is also a fundamental matrix.

Complementary System

The complementary system refers to the homogeneous part of the differential equation system from which particular solutions are derived. Essentially, it is the system obtained by removing any non-homogeneous terms (like constant vectors or functions that are added to the derivative terms).
Finding the solution to the complementary system involves solving for the kernel of the associated differential operator. This involves derivatives set equal to zero, which can often be solved by determining the eigenvalues and eigenvectors when dealing with constant coefficient systems.
The solutions to the complementary system form what is called the homogeneous solution. When combined with a particular solution, they constitute the full solution to the original non-homogeneous differential equation.

Matrix Inversion

Matrix inversion is a mathematical operation that finds the matrix \( A^{-1} \) such that when it is multiplied with the original matrix \( A \), it yields the identity matrix. In the context of differential equations, especially when dealing with a fundamental matrix, it is often necessary to find the inverse to solve for particular solutions.
The formula for the inverse of a 2x2 matrix \( \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \) is \( \frac{1}{ad - bc} \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right] \). The denominator \( ad - bc \) is the determinant. If the determinant is zero, the matrix is not invertible.
Matrix inversion is crucial in systems where we use the inverse of the fundamental matrix to express solutions to certain integral or algebraic equations. However, the existence of the matrix inverse hinges on the condition that the determinant is non-zero, ensuring the matrix is non-singular.