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 $$ \begin{aligned} x_{1}^{\prime} &=x_{2} \\ x_{2}^{\prime} &=-q(t) x_{1}-p(t) x_{2} \end{aligned} $$ Show that if \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\) are a fundamental set of solutions of Eqs. (ii), and if \(y^{(1)}\) and \(y^{(2)}\) are a fundamental set of solutions of Eq. (i), 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

Question: Show that the Wronskian of the second-order equation \(y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0\) is equal to a nonzero constant times the Wronskian of the system \(\begin{aligned} x_{1}^{\prime} &= x_{2} \\ x_{2}^{\prime} &=-q(t) x_{1}-p(t) x_{2} \end{aligned}\), given that \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\) are a fundamental set of solutions of the system, and \(y^{(1)}\) and \(y^{(2)}\) are a fundamental set of solutions of the second-order equation.

Step-by-step solution

Expressing y^{(1)} and y^{(2)} as linear combinations of x_{11} and x_{12}

According to the hint, we can write \(y^{(1)}(t)\) and \(y^{(2)}(t)\) as linear combinations of \(x_{11}(t)\) and \(x_{12}(t)\): $$ \begin{aligned} y^{(1)}(t) &= a_{1} x_{11}(t) + a_{2} x_{12}(t) \\ y^{(2)}(t) &= b_{1} x_{11}(t) + b_{2} x_{12}(t) \end{aligned} $$ where \(a_1, a_2, b_1,\) and \(b_2\) are constants.

Finding the Wronskians of the second-order equation and the system

Now, let's find the Wronskian for the second-order equation, \(W\left[y^{(1)}, y^{(2)}\right]\), which is given by: $$ W\left[y^{(1)}, y^{(2)}\right] = \begin{vmatrix} y^{(1)} & y^{(2)} \\ y^{(1) \prime} & y^{(2) \prime} \end{vmatrix}= y^{(1)}y^{(2) \prime} - y^{(1) \prime} y^{(2)} $$ Similarly, we find the Wronskian for the system, \(W\left[\mathbf{x}^{(1)}, \mathbf{x}^{(2)}\right]\), as: $$ W\left[\mathbf{x}^{(1)}, \mathbf{x}^{(2)}\right] = \begin{vmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{vmatrix}= x_{11}x_{22} - x_{21}x_{12} $$

Relating the Wronskians

Now, we need to show that the Wronskians are related by the equation \(W\left[y^{(1)}, y^{(2)}\right]=c W\left[\mathbf{x}^{(1)}, \mathbf{x}^{(2)}\right]\), where \(c\) is a nonzero constant. We express the Wronskian for the second-order equation in terms of \(x_{11}\) and \(x_{12}\) using the linear combinations derived in Step 1: $$ W\left[y^{(1)}, y^{(2)}\right] = (a_{1}x_{11} + a_{2}x_{12})(b_{1}x_{22} - b_{2}x_{21}) - (a_{1}x_{21} + a_{2}x_{22})(b_{1}x_{11} - b_{2}x_{12}) $$ $$ W\left[y^{(1)}, y^{(2)}\right] = a_{1}b_{1}(x_{11}x_{22} - x_{21}x_{12}) $$ Now we can see that \(W\left[y^{(1)}, y^{(2)}\right]\) is equal to the Wronskian of the system, \(W\left[\mathbf{x}^{(1)}, \mathbf{x}^{(2)}\right]\), times the constant \(c=a_{1}b_{1}\), and since \(a_1\) and \(b_1\) are constants, \(c\) is a nonzero constant. Therefore, the relationship between the Wronskians is proven: $$ W\left[y^{(1)}, y^{(2)}\right]=c W\left[\mathbf{x}^{(1)}, \mathbf{x}^{(2)}\right] $$

Key concepts

Second-Order Linear Equations

Second-order linear equations are differential equations of the form \(y'' + p(t)y' + q(t)y = 0 \). These equations involve the second derivative of an unknown function \(y\) with respect to a variable \(t\), as well as its first derivative and the function itself. A typical example would be found in physics, such as the damped harmonic oscillator equation.

Understanding and solving second-order linear equations is crucial because they describe a wide range of phenomena in science and engineering. Key properties of these equations include their linearity and constant coefficients, which allow us to apply superposition principles in finding their solutions.

One method of solving such equations involves the use of a system of first-order linear equations, which can simplify the analysis and solution process. By translating a second-order equation into a system, we can use powerful mathematical tools like matrix theory and the Wronskian to find solutions.

Wronskian

The Wronskian is a determinant used in the theory of differential equations to determine whether a set of solutions is linearly independent. For two functions, \(y^{(1)}\) and \(y^{(2)}\), the Wronskian is defined as:

\[ W(y^{(1)}, y^{(2)}) = y^{(1)}y^{(2)\prime} - y^{(1)\prime} y^{(2)} \]

This can be extended to more functions by using appropriately larger determinants. The Wronskian is vital because if it is nonzero at some point, it indicates that the functions \(y^{(1)}\) and \(y^{(2)}\) are linearly independent over some interval. Linear independence is crucial for forming a fundamental set of solutions to differential equations.

In the context of the original exercise, the Wronskian relates to both the second-order differential equation and the corresponding first-order system. Demonstrating the relationship between these Wronskians shows the equivalence of solutions across both representations of the problem.

Fundamental Set of Solutions

A fundamental set of solutions is a collection of solutions to a differential equation that forms a basis for the solution space of that equation. This means that any solution to the differential equation can be expressed as a linear combination of the solutions in the fundamental set.

For a second-order linear differential equation like \(y'' + p(t)y' + q(t)y = 0 \), a fundamental set of solutions would typically consist of two linearly independent solutions. The general solution can be written as \(c_1 y^{(1)}(t) + c_2 y^{(2)}(t)\), where \(c_1\) and \(c_2\) are constants.

In the exercise, the task is to show that solutions to a system of first-order equations can also form a fundamental set for the original second-order problem. This involves expressing the solutions of the original differential equation in terms of solutions from the derived system, thus providing a deep connection between different representations of differential problems.