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Chapter 4: Problem 26

The given functions are solutions of the homogeneous linear system. (a) Compute the Wronskian of the solution set and verify that the solution set is a fundamental set of solutions. (b) Compute the trace of the coefficient matrix. (c) Verify Abel's theorem by showing that, for the given point \(t_{0}\), $$ W(t)=W\left(t_{0}\right) e^{\int_{t_{0}}^{t} \operatorname{tr}(P(s)] d s} $$ $$ \mathbf{y}^{\prime}=\left[\begin{array}{cc} 1 & t \\ 0 & -t^{-1} \end{array}\right] \mathbf{y}, \quad t \neq 0 $$

Short Answer

Expert verified
b) What is the trace of the given coefficient matrix? c) Is it possible to directly verify Abel's theorem in this case?

Step by step solution

01

Finding the Wronskian

First, we need to compute the Wronskian of the given functions. The Wronskian is given by the determinant of the matrix formed by the functions and their derivatives. Since we are not provided with the functions themselves, we cannot compute the Wronskian explicitly. However, we can check if the Wronskian is non-zero for any point t 鈮� 0 to see if the functions form a fundamental set of solutions. Let the two functions be \(y_1(t)\) and \(y_2(t)\). The Wronskian W(t) is then given by: $$ W(t) = \det \begin{bmatrix} y_1(t) & y_2(t) \\ y_1'(t) & y_2'(t) \end{bmatrix} $$ If \(W(t) \neq 0\) for any \(t \neq 0\), the set of functions is a fundamental set of solutions.
02

Computing the trace of the coefficient matrix

The trace of a matrix is the sum of its diagonal elements. The given coefficient matrix is: $$ \begin{bmatrix}1 & t \\ 0 & -t^{-1}\end{bmatrix} $$ The trace of this matrix is: $$ \operatorname{tr}(P(t)) = 1 - t^{-1} $$
03

Verifying Abel's theorem

Abel's theorem states that: $$ W(t) = W(t_{0}) e^{\int_{t_{0}}^{t} \operatorname{tr}(P(s)) ds} $$ We have already calculated the trace of the coefficient matrix as \(1 - t^{-1}\). Now, we need to compute the integral of the trace and check if Abel's theorem holds true. To do this, we will first compute the integral: $$ \int_{t_0}^t (1 - t^{-1}) dt = \int_{t_0}^t dt - \int_{t_0}^t t^{-1} dt \\ = (t - t_0) - (\ln |t| - \ln |t_0|) = (t - t_0) - \ln \frac{t}{t_0} $$ With this, Abel's theorem becomes: $$ W(t) = W(t_{0}) e^{(t - t_0) - \ln \frac{t}{t_0}} $$ Since we cannot compute \(W(t)\) explicitly, we cannot verify Abel's theorem directly. However, if the Wronskian is nonzero for any point \(t \neq 0\), then Abel's theorem will hold true. Thus, if the functions form a fundamental set of solutions, Abel's theorem will be verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fundamental Set of Solutions
A fundamental set of solutions is a critical concept in differential equations. It refers to a set of solutions to a homogeneous linear differential equation that can be used to express all other solutions. This concept is based on linear independence. For a set of solutions to be fundamental, their Wronskian must be non-zero for any point in the interval of interest.

The Wronskian is a determinant that helps determine if a collection of functions is linearly independent. If given functions form the columns of a matrix, the Wronskian is the determinant of that matrix. For two functions, say, \(y_1(t)\) and \(y_2(t)\), it's given by:
  • \(W(t) = \det \begin{bmatrix} y_1(t) & y_2(t) \ y_1'(t) & y_2'(t) \end{bmatrix}\)
Therefore, if the Wronskian \(W(t) eq 0\) for any \(t eq 0\), the solution set is a fundamental set of solutions and serves as a basis for all solutions of the homogeneous linear differential equation.
Trace of a Matrix
The trace of a matrix is a straightforward concept: it is simply the sum of the diagonal elements of a square matrix. This mathematical operation captures certain properties of matrices that are useful in various applications.

In the context of our differential equation, the coefficient matrix is:
  • \(\begin{bmatrix} 1 & t \ 0 & -t^{-1} \end{bmatrix}\)
The trace can be calculated by summing the diagonal elements:
  • \(\text{tr}(P(t)) = 1 - t^{-1}\)
The trace has significance in solving the differential equation by contributing to the determination of stability and properties related to the eigenvalues of the matrix. Knowing the trace helps in utilizing established theorems like Abel's theorem, which relates the trace to the behavior of the solutions.
Abel's Theorem
Abel's theorem is a result in the theory of linear differential equations that expresses the relationship between the Wronskian of a solution set and the trace of the coefficient matrix. It states that for a system of linear differential equations, the Wronskian changes according to an exponential function depending on the trace.

Formally, Abel's theorem is given as:
  • \(W(t) = W(t_{0}) e^{\int_{t_{0}}^{t} \text{tr}(P(s)) \ ds}\)
Here, \(\text{tr}(P(s))\) is the trace of the coefficient matrix \(P(s)\). The integral \(\int_{t_0}^t \text{tr}(P(s)) \ ds\) captures the accumulated effect of the trace from \(t_0\) to \(t\).

By substituting the calculated trace, we confirm whether the solutions adhere to Abel's theorem. In our case, the expression resolves to:
  • \(W(t) = W(t_0) e^{(t - t_0) - \ln \frac{t}{t_0}}\)
Although we cannot compute the Wronskian explicitly due to the lack of specific functions, verifying the non-zero nature of the Wronskian suffices to confirm Abel's theorem, ensuring that solutions remain valid across the specified system.

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Most popular questions from this chapter

A Spring-Mass-Dashpot System with Variable Damping As we saw in Section \(3.6\), the differential equation modeling unforced damped motion of a mass suspended from a spring is \(m y^{\prime \prime}+\gamma y^{\prime}+k y=0\), where \(y(t)\) represents the downward displacement of the mass from its equilibrium position. Assume a mass \(m=1 \mathrm{~kg}\) and a spring constant \(k=4 \pi^{2} \mathrm{~N} / \mathrm{m}\). Also assume the damping coefficient \(\gamma\) is varying with time: $$ \gamma(t)=2 t e^{-t / 2} \mathrm{~kg} / \mathrm{sec} \text {. } $$ Assume, at time \(t=0\), the mass is pulled down \(20 \mathrm{~cm}\) and released. (a) Formulate the appropriate initial value problem for the second order scalar differential equation, and rewrite it as an equivalent initial value problem for a first order linear system. (b) Applying Euler's method, numerically solve this problem on the interval \(0 \leq t \leq 10 \mathrm{~min}\). Use a step size of \(h=0.005\). (c) Plot the numerical solution on the time interval \(0 \leq t \leq 10 \mathrm{~min}\). Explain, in qualitative terms, the effect of the variable damping upon the solution.

Find the largest interval \(a

Each initial value problem was obtained from an initial value problem for a higher order scalar differential equation. What is the corresponding scalar initial value problem? $$ \mathbf{y}^{\prime}=\left[\begin{array}{rr} 0 & 1 \\ -3 & 2 \end{array}\right] \mathbf{y}+\left[\begin{array}{c} 0 \\ 2 \cos 2 t \end{array}\right], \quad \mathbf{y}(-1)=\left[\begin{array}{l} 1 \\ 4 \end{array}\right] $$

Each of the systems of linear differential equations can be expressed in the form \(\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t) .\) Determine \(P(t)\) and \(\mathbf{g}(t)\) $$ A^{\prime \prime}(t)=\left[\begin{array}{ll} 1 & t \\ 0 & 0 \end{array}\right], \quad A(0)=\left[\begin{array}{rr} 1 & 1 \\ -2 & 1 \end{array}\right], \quad A(1)=\left[\begin{array}{ll} -1 & 2 \\ -2 & 3 \end{array}\right] $$

In each exercise, determine all equilibrium solutions (if any).\(\mathbf{y}^{\prime}=\left[\begin{array}{rr}1 & -1 \\ -1 & 1\end{array}\right] \mathbf{y}+\left[\begin{array}{r}2 \\\ -2\end{array}\right]\)
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