Question

Let \(A\) be a lower triangular \(n \times n\) matrix. If \(\Phi\) is a solution of \(Y^{\prime}=A Y\), then $$ \phi_{i}(t)=\sum_{i=1}^{i} p_{i j}(t) e^{a_{1} t}, \quad i=1,2, \ldots, n $$ where \(p_{i j}(t)\) are polynomials.

Short answer

Short answer: The given formula for the solution of a first-order linear homogeneous system of ODEs with constant coefficients and a lower triangular n×n matrix A is a linear combination of exponential functions with polynomial coefficients. To demonstrate its validity, we started from the system of ODEs, Y'(t) = AY(t), and analyzed the given lower triangular matrix A. Then, we rewrote the system using A and derived the formula for each component of the solution vector Y(t) as a linear combination of exponential functions with polynomial coefficients. After finding the derivatives of the solution components and substituting them back into the system, we showed that the given formula holds for a lower triangular matrix A.

Step-by-step solution

Write the linear system of ODEs

For a given matrix A, we have the following system of ODEs: $$ Y'(t) = AY(t) $$ where Y(t) is a vector of n unknown functions to be determined, and Y'(t) is the vector of their derivatives with respect to t.

Analyze the lower triangular matrix A

Because A is a lower triangular matrix, all elements in the upper right part of the matrix are zero. This means that the ith row of A has non-zero elements only in the first i columns, and zero elements elsewhere: $$ A = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ a_{21} & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} $$

Rewrite the system in terms of the lower triangular matrix A

Substituting the lower triangular matrix A into the given system of ODEs, we have: $$ \begin{bmatrix} \phi_1'(t) \\ \phi_2'(t) \\ \vdots \\ \phi_n'(t) \end{bmatrix} = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ a_{21} & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \begin{bmatrix} \phi_1(t) \\ \phi_2(t) \\ \vdots \\ \phi_n(t) \end{bmatrix} $$

Derive the given formula for the solution

For each component \(\phi_i(t)\) of the solution vector Y(t), we rewrite the component as a linear combination of exponential functions with polynomial coefficients \(p_{ij}(t)\), as given by the formula: $$ \phi_i(t) = \sum_{j=1}^i p_{ij}(t)e^{a_{j1}t} $$ The sum is taken from j=1 to j=i because the matrix A is lower triangular, and so, the ith equation in the system involves only \(\phi_j\) with j≤i.

Find the derivatives of the solution components

Now we need to find \(\phi_i'(t)\) for each i=1,2,...,n. Using the product rule and the chain rule, we get: $$ \phi_i'(t) = \sum_{j=1}^i (p_{ij}'(t)e^{a_{j1}t} + p_{ij}(t)a_{j1}e^{a_{j1}t}) $$

Substitute the solution components and their derivatives back into the system

Now we substitute the expressions for \(\phi_i(t)\) and \(\phi_i'(t)\) into the system from Step 3, which gives us: $$ \begin{bmatrix} \sum_{j=1}^1 (p_{1j}'(t)e^{a_{j1}t} + p_{1j}(t)a_{j1}e^{a_{j1}t}) \\ \sum_{j=1}^2 (p_{2j}'(t)e^{a_{j1}t} + p_{2j}(t)a_{j1}e^{a_{j1}t}) \\ \vdots \\ \sum_{j=1}^n (p_{nj}'(t)e^{a_{j1}t} + p_{nj}(t)a_{j1}e^{a_{j1}t}) \end{bmatrix} = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ a_{21} & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \begin{bmatrix} \sum_{j=1}^1 p_{1j}(t)e^{a_{j1}t} \\ \sum_{j=1}^2 p_{2j}(t)e^{a_{j1}t} \\ \vdots \\ \sum_{j=1}^n p_{nj}(t)e^{a_{j1}t} \end{bmatrix} $$ This shows that the given formula for the solution vector Y(t) in terms of exponential functions with polynomial coefficients holds for a lower triangular matrix A, thus solving the exercise.

Key concepts

Lower Triangular Matrix

A lower triangular matrix is a special type of square matrix where all the elements above the diagonal are zeros. This characteristic simplifies complexities when solving linear systems involving this kind of matrix. Lower triangular matrices have the form: \[ A = \begin{bmatrix}a_{11} & 0 & \cdots & 0 \a_{21} & a_{22} & \cdots & 0 \\vdots & \vdots & \ddots & \vdots \a_{n1} & a_{n2} & \cdots & a_{nn}\end{bmatrix}\]Several properties make these matrices important:

• Efficiency in solving linear equations: Systems using lower triangular matrices can be solved using forward substitution, a method faster than other techniques required for more general matrices.
• Unique solutions: When the diagonal entries are non-zero, the set of equations associated with the system is guaranteed to have a unique solution.
• Stability: Operations involving lower triangular matrices tend to preserve stability, making them ideal in numerical applications.

These features are crucial when solving differential equations such as those described in the exercise.

Linear System of ODEs

A linear system of ordinary differential equations (ODEs) comprises several first-order linear differential equations involving multiple variables. Each equation in the system can be expressed in the form: \[ Y'(t) = AY(t) \]where:

• \(Y(t)\) is a vector containing the unknown functions we aim to solve for.
• \(Y'(t)\) represents their derivatives.
• \(A\) is a matrix that connects each equation in the system.

The objective is to find a vector-function \(\Phi\) that satisfies this equation for all \(t\). In our exercise, since \(A\) is a lower triangular matrix, the linear system is simpler to solve using successive substitution techniques.
Linear ODE systems appear in various science and engineering fields, such as modeling electrical circuits or describing population dynamics. The understanding of how matrices like \(A\) interact within these systems provides profound insights into these applications.

Solution of ODEs with Polynomial Coefficients

The solution of ODEs with polynomial coefficients often involves a combination of polynomial terms and exponential functions. The general form of the solution we've seen in the exercise can be described as: \[ \phi_i(t) = \sum_{j=1}^i p_{ij}(t)e^{a_{j1}t} \]where:

• \(\phi_i(t)\) is the solution component associated with the \(i\)-th equation of the ODE system.
• \(p_{ij}(t)\) are polynomials representing coefficients of the exponential functions.
• The exponential terms \(e^{a_{j1}t}\) appear due to the derivative's nature of exponential functions.

The sum from \(j=1\) to \(i\) ensures that each solution component \(\phi_i(t)\) only depends on itself and the preceding components, aligning with the structure of the lower triangular matrix. This approach simplifies the resolution by allowing step-by-step solving, starting from the first equation to the nth equation, progressively reducing complexity.
Mastering this technique is crucial for tackling differential equations with non-constant coefficients, commonly found in real-world phenomena with variable conditions over time.