Question

If \(x_{1}=y\) and \(x_{2}=y^{\prime},\) then the second-order equation $$y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0$$ corresponds to the system \\[ \begin{array}{l} x_{1}^{\prime}=x_{2} \\ x_{2}^{\prime}=-q(t) x_{1}-p(t) x_{2} \end{array} \\] Show that if \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\) are a fundamental set of solutions of equations (19), and if \(y^{(1)}\) and \(y^{(2)}\) are a fundamental set of solutions of equation \((18),\) then \(W\left[y^{(1)}, y^{(2)}\right]=c W\left[\mathbf{x}^{(1)}, \mathbf{x}^{(2)}\right],\) where \(c\) is a nonzero constant. Hint: \(y^{(1)}(t)\) and \(y^{(2)}(t)\) must be linear combinations of \(x_{11}(t)\) and \(x_{12}(t)\)

Short answer

In this exercise, we have shown that the Wronskians for both the second-order equation and the system of equations are proportional by a non-zero constant. This conclusion comes from expressing the solutions of the second-order equation as linear combinations of the solutions of the system of equations, and computing their respective Wronskians. Comparing the expressions for the Wronskians, we could identify a non-zero constant relating them, satisfying the requirement for proportionality.

Step-by-step solution

Relate Solutions of both Systems

Given that \(x_1=y\) and \(x_2=y'\), let's represent the fundamental solutions for both systems in the following way: - For the second-order equation \((18)\), the solutions are denoted by \(y^{(1)}\) and \(y^{(2)}\). - For the system of equations \((19)\), the solutions are represented by vectors \(\mathbf{x}^{(1)} = \begin{bmatrix} x_{11} \\ x_{21} \end{bmatrix}\) and \(\mathbf{x}^{(2)} = \begin{bmatrix} x_{12} \\ x_{22} \end{bmatrix}\). Now, as given in the hint, we can express \(y^{(1)}(t)\) and \(y^{(2)}(t)\) as linear combinations of \(x_{11}(t)\) and \(x_{12}(t)\). Let: $$y^{(1)}(t)=A_{11}x_{11}(t)+A_{12}x_{12}(t)$$ $$y^{(2)}(t)=A_{21}x_{11}(t)+A_{22}x_{12}(t)$$

Compute the Wronskian of Solutions of the Second-Order Equation

We will now compute the Wronskian \(W\left[y^{(1)},y^{(2)}\right]\): $$W\left[y^{(1)},y^{(2)}\right] = y^{(1)}y^{'(2)} - y^{(2)}y^{'(1)}$$ By substituting the expressions from step 1 and noting that \(x_2 = y'\), we get: \begin{align*} W\left[y^{(1)},y^{(2)}\right] &= (A_{11}x_{11}+A_{12}x_{12})(x_{21}A_{21}+x_{22}A_{22}) - (A_{21}x_{11}+A_{22}x_{12})(x_{21}A_{11}+x_{22}A_{12}) \\ &= (A_{11}A_{21}-A_{21}A_{11})x_{11}x_{21} + (A_{11}A_{22}-A_{12}A_{21})x_{11}x_{22} \\ &\quad + (A_{12}A_{21}-A_{21}A_{12})x_{12}x_{21} + (A_{12}A_{22}-A_{22}A_{12})x_{12}x_{22} \end{align*}

Compute the Wronskian of Solutions of the System of Equations

Now, let's compute the Wronskian \(W\left[\mathbf{x}^{(1)},\mathbf{x}^{(2)}\right]\): $$W\left[\mathbf{x}^{(1)},\mathbf{x}^{(2)}\right]=\det\begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix} = x_{11}x_{22} - x_{12}x_{21}$$

Prove the Proportionality of Wronskians

Comparing the Wronskians we calculated in steps 2 and 3: $$W\left[y^{(1)},y^{(2)}\right] = (A_{11}A_{21}-A_{21}A_{11})x_{11}x_{21} + (A_{11}A_{22}-A_{12}A_{21})x_{11}x_{22} \\ \qquad + (A_{12}A_{21}-A_{21}A_{12})x_{12}x_{21} + (A_{12}A_{22}-A_{22}A_{12})x_{12}x_{22}$$ $$W\left[\mathbf{x}^{(1)},\mathbf{x}^{(2)}\right]=x_{11}x_{22} - x_{12}x_{21}$$ Observe that \(W\left[y^{(1)},y^{(2)}\right]\) can be written as: $$W\left[y^{(1)},y^{(2)}\right] = c\left(x_{11}x_{22}-x_{12}x_{21}\right) = cW\left[\mathbf{x}^{(1)},\mathbf{x}^{(2)}\right]$$ Where \(c = (A_{11}A_{21}-A_{21}A_{11}) + (A_{11}A_{22}-A_{12}A_{21}) + (A_{12}A_{21}-A_{21}A_{12}) + (A_{12}A_{22}-A_{22}A_{12})\). Hence, \(W\left[y^{(1)},y^{(2)}\right]=c W\left[\mathbf{x}^{(1)},\mathbf{x}^{(2)}\right]\) with a non-zero constant \(c\). This proves the proportionality of Wronskians for both systems of equations.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They are central in modeling a vast array of physical phenomena, from the motion of celestial bodies to the behavior of electronic circuits. A differential equation expresses the rate at which some quantity changes.

Differential equations can be categorized by their order, determined by the highest derivative in the equation. For instance, a second-order differential equation will involve the second derivative of the function. Solving these equations typically yields a function or a set of functions that satisfy the given relationship between the derivatives and the function itself.

Linear Independence

Linear independence is a fundamental concept in algebra, particularly in the study of vector spaces and solutions to differential equations. A set of functions (or vectors) is said to be linearly independent if no function (or vector) in the set can be written as a linear combination of the others.

For a set of functions to be a fundamental set of solutions to a differential equation, the functions must be linearly independent. This ensures that the solutions span the entire solution space, allowing for the expression of any solution to the differential equation as a linear combination of these fundamental solutions.

System of Equations

A system of equations is a collection of two or more equations involving a number of variables. The solutions to a system of equations are the values that satisfy all the equations simultaneously. Systems can have a single unique solution, infinitely many solutions, or no solution at all.

In the context of differential equations, systems often arise when looking to convert higher-order differential equations into a set of first-order equations. This enables us to analyze complex dynamic systems by examining the interrelationships between multiple simultaneous changes. The technique of converting a second-order differential equation into a system of first-order equations, as shown in the original exercise, is invaluable in such analyses.

Second-order Linear Differential Equation

A second-order linear differential equation is characterized by the presence of a second derivative of an unknown function, without any powers or products of the function and its derivatives. It is of the form:\[ a(t)y'' + b(t)y' + c(t)y = 0 \]where the coefficients \(a(t)\), \(b(t)\), and \(c(t)\) are functions of the independent variable \(t\) but do not depend on the unknown function \(y\) or its derivatives. This equation may represent physical systems like oscillations, electrical circuits, or mechanical movements.

The original exercise demonstrates a case of a second-order differential equation being rewritten as a system of first-order differential equations. A set of solutions to this equation, if linearly independent, forms a basis for the solution space, enabling us to describe the general behavior of systems governed by such equations.