Question

In Section 9.1 the Wronskian of \(n\) solutions \(y_{1}, y_{2}, \ldots, y_{n}\) of the \(n-\) th order equation $$ P_{0}(x) y^{(n)}+P_{1}(x) y^{(n-1)}+\cdots+P_{n}(x) y=0 $$ was defined to be $$ W=\left|\begin{array}{cccc} y_{1} & y_{2} & \cdots & y_{n} \\ y_{1} & y_{2}^{\prime} & \cdots & y_{n}^{\prime} \\ \vdots & \vdots & \ddots & \vdots \\ y_{1}^{(n-1)} & y_{2}^{(n-1)} & \cdots & y_{n}^{(n-1)} \end{array}\right| . $$ (a) Rewrite (A) as a system of first order equations and show that \(W\) is the Wronskian (as defined in this section) of \(n\) solutions of this system. (b) Apply Eqn. ( 10.3 .6 ) to the system derived in (a), and show that $$ W(x)=W\left(x_{0}\right) \exp \left\\{-\int_{x_{0}}^{x} \frac{P_{1}(s)}{P_{0}(s)} d s\right\\} $$ which is the form of Abel's formula given in Theorem \(9.1 .3 .\)

Short answer

Question: Given an nth-order linear differential equation and the Wronskian of its n linearly independent solutions, show that the Wronskian of a system of first-order linear differential equations derived from the nth-order equation is the same as the given Wronskian and find a formula for that Wronskian. Answer: The Wronskian of the system of first-order linear differential equations derived from the nth-order equation is the same as the Wronskian of the original equation. The formula for the Wronskian is given by: $$ W(x) = W(x_0) \exp{\left(-\int_{x_{0}}^{x}\frac{P_{1}(s)}{P_{0}(s)} ds\right)}. $$

Step-by-step solution

Part (a): Rewrite the nth order equation as a system of first-order equations

In order to rewrite the nth order linear differential equation as a system of first-order equations, let's define new variables: $$ z_{1}(x) = y(x), \\ z_{2}(x) = y'(x), \\ z_{3}(x) = y''(x), \\ \vdots \\ z_{n}(x) = y^{(n-1)}(x). $$ Now the given nth order linear differential equation can be written as a set of n first-order linear differential equations: $$ z'_{1}(x) = z_{2}(x), \\ z'_{2}(x) = z_{3}(x), \\ z'_{3}(x) = z_{4}(x), \\ \vdots \\ z'_{n-1}(x) = z_{n}(x), \\ z'_{n}(x) = -\frac{P_{n}(x)}{P_{0}(x)}z_{1}(x) - \frac{P_{n-1}(x)}{P_{0}(x)}z_{2}(x) - \cdots - \frac{P_{1}(x)}{P_{0}(x)}z_{n}(x). $$ Now, we need to show that W is the Wronskian of n solutions of this system. The Wronskian of the system is the determinant of the matrix whose columns are the n solutions \(\textbf{z}_{i}(x) = (z_{1i}, z_{2i}, \ldots, z_{ni})^T\) such that \(z_{ij}^{(k)}(x)=y_{j}^{(k)}(x)\) for \(k=0,1,2,\ldots,n-1\). The matrix will be $$ \begin{pmatrix} y_{1} & y_{2} & \cdots & y_{n} \\ y_{1}' & y_{2}' & \cdots & y_{n}' \\ \vdots & \vdots & \ddots & \vdots \\ y_{1}^{(n-1)} & y_{2}^{(n-1)} & \cdots & y_{n}^{(n-1)} \end{pmatrix}. $$ So, the Wronskian of the system is the same as the Wronskian defined in Section 9.1.

Part (b): Apply Eqn. ( 10.3 .6 ) to the first-order system

First, let's recall Eqn. (10.3.6): $$ \frac{dW}{dx} = -\frac{P_{1}(x)}{P_{0}(x)}W(x). $$ We will now apply this equation to our system of first-order linear differential equations derived in part (a). To determine the W(x) for this system, we need to integrate both sides of the equation with respect to x: $$ \int\frac{dW}{W(x)} = -\int\frac{P_{1}(s)}{P_{0}(s)} ds. $$ Integrating both sides, we have $$ \ln(W(x)) = -\int\frac{P_{1}(s)}{P_{0}(s)} ds + \ln(W(x_0)). $$ Finally, we can find the expression for W(x) by applying the exponential function to both sides: $$ W(x) = W(x_0) \exp{\left(-\int_{x_{0}}^{x}\frac{P_{1}(s)}{P_{0}(s)} ds\right)}. $$ This result is the form of Abel's formula given in Theorem 9.1.3.

Key concepts

System of First-Order Differential Equations

When we dive into the world of differential equations, a system of first-order differential equations is a fundamental concept that can simplify many complicated scenarios. Imagine breaking down a complex problem into smaller, manageable pieces that interact with one another. This analogously is what occurs when we convert higher-order differential equations into systems of first-order ones.

We begin by introducing new variables to represent each derivative of our original function up to the \(n-1\)th derivative. For instance, \(y\)'s first derivative becomes \(z_2\), its second derivative becomes \(z_3\), and so forth, up to \(z_n\). Each \(z_i\) is related to \(z_{i+1}\) with \(z'_i = z_{i+1}\), creating a sequence of first-order equations. The last equation, however, relates \(z'_{n}\) back to all the previous \(z_i\)'s, with coefficients determined by the original nth order equation terms.

In an exam or a homework problem, being able to construct this system from a higher-order equation not only shows understanding of the mechanics but also sets the stage for analyzing the solution's behavior using methods specific to first-order systems. For students aiming to master these conversions, practicing with various differential equations will reveal patterns and methods to streamline the process.

Abel's Formula

Step into the world of solutions for linear differential equations with the elegant Abel's formula. This gem of differential equations is often used to assess the nature of the solutions without explicitly solving the equation. Abel's formula provides a direct relationship between the Wronskian determinant, which is a measure of the linear independence of a set of functions, and the coefficients of the differential equation.

To utilize Abel's formula, imagine having a ribbon and measuring how it twists and turns over a certain path. This is similar to the way we track how the Wronskian evolves as we move along the \(x\)-axis. Abel's formula states that the logarithm of the Wronskian is the negative integral of the ratio of the first two coefficients of the differential equation. When we exponentiate both sides, we find that the Wronskian at any point \(x\) can be found by multiplying the Wronskian at another point \(x_0\) by an exponential factor, which accounts for how the coefficients \(P_{1}(s) / P_{0}(s)\) accumulate effect as \(s\) goes from \(x_0\) to \(x\).

This relationship becomes particularly helpful when determining if a set of solutions forms a fundamental set, which is a critical step in solving linear differential equations. The exercise on Abel's formula is a testament to its utility, showing that even without direct solutions, powerful conclusions about differential equations can be drawn.

Nth Order Linear Differential Equation

At the heart of many mathematical and physical systems lies the nth order linear differential equation, an equation that ties together derivatives of various orders of a function. These versatile equations can model phenomena ranging from the vibrations of a string to the behavior of electrical circuits.

In its most general form, an nth order linear differential equation looks like a towering stack of terms, each involving a derivative of the function we're trying to uncover, coupled with a coefficient function dependent on \(x\). The equation stipulates that when all these terms are summed, they cancel out to be zero. This is a homogenous linear differential equation and understanding its structure is essential. It tells us that the function's future behavior is determined by a linear combination of its current and past rates of change.

One reason students might find nth order linear differential equations intimidating is because of their complexity; however, learning methods like transforming them into systems of first-order equations streamlines the thought process. The powerful tool of the Wronskian helps in studying the solutions' independence and is a cornerstone in theories like the superposition principle of linear systems. Thus, studying nth order linear differential equations often involves breaking the problem into smaller pieces, using the Wronskian, and applying key formulas like Abel's to explore the intricate dance of their solutions.