Question

Suppose $$ \mathbf{y}_{1}=\left[\begin{array}{l} y_{11} \\ y_{21} \end{array}\right] \quad \text { and } \quad \mathbf{y}_{2}=\left[\begin{array}{l} y_{12} \\ y_{22} \end{array}\right] $$ are solutions of the \(2 \times 2\) system \(\mathbf{y}^{\prime}=A \mathbf{y}\) on \((a, b),\) and let $$ Y=\left[\begin{array}{ll} y_{11} & y_{12} \\ y_{21} & y_{22} \end{array}\right] \quad \text { and } \quad W=\left|\begin{array}{ll} y_{11} & y_{12} \\ y_{21} & y_{22} \end{array}\right| $$ thus, \(W\) is the Wronskian of \(\left\\{\mathbf{y}_{1}, \mathbf{y}_{2}\right\\}\). (a) Deduce from the definition of determinant that $$ W^{\prime}=\left|\begin{array}{ll} y_{11}^{\prime} & y_{12}^{\prime} \\ y_{21} & y_{22} \end{array}\right|+\left|\begin{array}{ll} y_{11} & y_{12} \\ y_{21}^{\prime} & y_{22}^{\prime} \end{array}\right| $$ (b) Use the equation \(Y^{\prime}=A(t) Y\) and the definition of matrix multiplication to show that $$ \left[\begin{array}{ll} y_{11}^{\prime} & y_{12}^{\prime} \end{array}\right]=a_{11}\left[\begin{array}{ll} y_{11} & y_{12} \end{array}\right]+a_{12}\left[y_{21} \quad y_{22}\right] $$ and $$ \begin{array}{ll} {\left[y_{21}^{\prime}\right.} & \left.y_{22}^{\prime}\right]=a_{21}\left[y_{11}\right. & \left.y_{12}\right]+a_{22}\left[y_{21}\right. & \left.y_{22}\right] \end{array} $$ (c) Use properties of determinants to deduce from (a) and (a) that $$ \left|\begin{array}{ll} y_{11}^{\prime} & y_{12}^{\prime} \\ y_{21} & y_{22} \end{array}\right|=a_{11} W \quad \text { and } \quad\left|\begin{array}{ll} y_{11} & y_{12} \\ y_{21}^{\prime} & y_{22}^{\prime} \end{array}\right|=a_{22} W $$ (d) Conclude from (c) that $$ W^{\prime}=\left(a_{11}+a_{22}\right) W $$ and use this to show that if \(a

Short answer

Question: Given a \(2 \times 2\) differential system with a constant matrix A and two solutions, find the Wronskian of the system. Answer: The Wronskian W(t) is given by the equation: $$ W(t) = W(t_0) \exp\left(\int_{t_0}^{t}\left[a_{11}(s) + a_{22}(s)\right]ds\right) \quad a<t<b $$ where \(a_{11}(s)\) and \(a_{22}(s)\) are the coefficients of matrix A, and \(W(t_0)\) is the Wronskian evaluated at \(t_0\).

Step-by-step solution

Derivative of the Wronskian

Using the definition of determinant, we have: $$ W' = \frac{d}{dt}\left|\begin{array}{ll} y_{11} & y_{12} \\\ y_{21} & y_{22} \end{array}\right| = \left|\begin{array}{ll} y_{11}^{\prime} & y_{12}^{\prime} \\\ y_{21} & y_{22} \end{array}\right|+\left|\begin{array}{ll} y_{11} & y_{12} \\\ y_{21}^{\prime} & y_{22}^{\prime} \end{array}\right| $$

Show relationship between derivatives of the solutions and matrix A coefficients

We are given \(Y^{\prime} = A(t)Y\). Therefore: $$ \left[\begin{array}{ll} y_{11}^{\prime} & y_{12}^{\prime} \end{array}\right]=a_{11}\left[\begin{array}{ll} y_{11} & y_{12} \end{array}\right]+a_{12}\left[y_{21} \quad y_{22}\right] $$ and $$ \begin{array}{ll} {\left[y_{21}^{\prime}\right.} & \left.y_{22}^{\prime}\right]=a_{21}\left[y_{11}\right. & \left.y_{12}\right]+a_{22}\left[y_{21}\right. & \left.y_{22}\right] \end{array} $$

Use properties of determinants to show relationship with matrix A coefficients

Using the properties of determinants and combining the results from Steps 1 and 2, we can deduce that: $$ \left|\begin{array}{ll} y_{11}^{\prime} & y_{12}^{\prime} \\\ y_{21} & y_{22} \end{array}\right| = a_{11}W $$ and $$ \left|\begin{array}{ll} y_{11} & y_{12} \\\ y_{21}^{\prime} & y_{22}^{\prime} \end{array}\right| = a_{22}W $$

Conclude with the specific form for the Wronskian

Using the results from Step 3, we can write the derivative of the Wronskian as: $$ W^{\prime} = (a_{11} + a_{22})W $$ To find \(W(t)\), we can now solve this first order linear ordinary differential equation: $$ \frac{dW}{dt} = (a_{11}(t) + a_{22}(t))W(t) $$ If \(a<t_0<b\), the solution to the above equation is: $$ W(t) = W(t_0) \exp\left(\int_{t_0}^{t}\left[a_{11}(s) + a_{22}(s)\right]ds\right) \quad a<t<b $$ This is the required result for Wronskian \(W(t)\).

Key concepts

System of Linear Differential Equations

Linear differential equations play a critical role in mathematics, especially when it comes to modelling real-world phenomena. A system of linear differential equations consists of several equations that can be compactly written in matrix form as \( \mathbf{y}^\prime = A \mathbf{y} \), where \( \mathbf{y} \) represents the vector of dependent variables and \( A \) is a matrix of coefficients.

Such systems arise naturally in many contexts, from engineering to economics, whenever quantities change at rates that depend on the quantities themselves. Solving these systems often involves finding an expression for \( \mathbf{y}(t) \), which describes the behavior of the system over time. Critical to this solution is the concept of the Wronskian determinant, which helps determine whether a set of solutions is linearly independent and can be used to measure the 'closeness' of solutions to singularity.

• To solve the system, one must find functions that satisfy all the equations simultaneously.
• Techniques such as eigenvalue methods or Laplace transforms can be utilized to find such solutions.
• Understanding the behavior of the Wronskian over an interval can offer insights into the stability and structure of solutions.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that allows us to compute the product of two matrices. It's crucial for representing linear transformations and is extensively used in systems of linear differential equations. The product of a matrix \( A \) and a vector \( \mathbf{y} \) is another vector where each entry is the sum of products of corresponding entries from the rows of \( A \) and columns of \( \mathbf{y} \).

For example, given a matrix \( A \) and a vector \( \mathbf{y} \), the multiplication \( A \mathbf{y} \) yields a new vector where:

• The i-th entry of the new vector is the dot product of the i-th row of \( A \) with \mathbf{y}\.
• Each entry represents a linear combination of the elements of \( \mathbf{y} \) with the corresponding row of matrix \( A \).

Understanding matrix multiplication is crucial for solving systems of differential equations because it demonstrates how the system evolves over time according to the rules defined by matrix \( A \).

Properties of Determinants

The determinant is a scalar value that can be computed from the elements of a square matrix and has numerous applications in linear algebra, calculus, and differential equations. Some key properties of determinants that aid in solving systems of linear differential equations include:

• Linearity: The determinant is linear in each row and column separately. This means if one row is the sum of two vectors, the determinant is the sum of the determinants of two matrices, identical except for having each of these vectors as that row.
• Product Rule: The determinant of the product of two matrices is the product of their determinants (\( |AB| = |A||B| \) if \( A \) and \( B \) are square matrices).
• Zero Determinant: A matrix with linearly dependent rows or columns has a determinant of zero.

The Wronskian, as a particular application of determinants, uses these properties when examining the solutions to differential equations. For instance, it can tell us if two solutions are independent by checking if the Wronskian is non-zero. When dealing with derivatives, the determinant can help us relate the derivative of a matrix of functions to the original matrix, as seen with the Wronskian's derivative in the exercise.