Question

In each exercise, (a) Rewrite the given \(n\)th order scalar initial value problem as \(\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}\), by defining \(y_{1}(t)=y(t), y_{2}(t)=y^{\prime}(t), \ldots, y_{n}(t)=y^{(n-1)}(t)\) and defining \(y_{1}(t)=y(t), y_{2}(t)=y(t), \ldots, y_{n}(t)=y^{\prime(n-t)}(t)\) \(\mathbf{y}(t)=\left[\begin{array}{c}y_{1}(t) \\ y_{2}(t) \\ \vdots \\\ y_{n}(t)\end{array}\right]\) (b) Compute the \(n^{2}\) partial derivatives \(\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{j}, i, j=1, \ldots, n\). (c) For the system obtained in part (a), determine where in \((n+1)\)-dimensional \(t \mathbf{y}\)-space the hypotheses of Theorem \(6.1\) are not satisfied. In other words, at what points \(\left(t, y_{1}, \ldots, y_{n}\right)\), if any, does at least one component function \(f_{i}\left(t, y_{1}, \ldots, y_{n}\right)\) and/or at least one partial derivative function \(\partial f_{i}\left(t, y_{1}, \ldots, y_{n}\right) / \partial y_{i}, i, j=1, \ldots, n\) fail to be continuous? What is the largest open rectangular region \(R\) where the hypotheses of Theorem \(6.1\) hold? $$ y^{\prime \prime}+t y^{\prime}+2 y=0, \quad y(0)=1, \quad y^{\prime}(0)=2 $$

Short answer

Answer: The theorem's hypotheses hold for all points in the (t, y_1, y_2, y_3) space since all the component functions and partial derivatives are continuous everywhere, and the largest open rectangular region R includes all points in the ℝ⁴ space.

Step-by-step solution

Define new variables

Let's define new variables to represent the function y(t) and its derivatives. $$ y_1(t) = y(t) \\ y_2(t) = y'(t) \\ y_3(t) = y''(t) $$

Rewrite the given ODE as a system of first-order ODEs

We have the given ODE as: $$ y''(t) + ty'(t) + 2y(t) = 0 $$ Now we substitute the variables from Step 1 so that we get a system of first-order ODEs. $$ y_3(t) + ty_2(t) + 2y_1(t) = 0 \\ $$ Differentiating the first two variables \(y_1(t)\) and \(y_2(t)\) with respect to t gives us: $$ y_1'(t) = y_2(t) \\ y_2'(t) = y_3(t) $$ We can express the system as follows: $$ y_1'(t) = y_2(t) \\ y_2'(t) = y_3(t) \\ y_3(t) = -ty_2(t) - 2y_1(t) \\ $$ With initial conditions: $$ \begin{cases} y_1(0) = 1 \\ y_2(0) = 2 \end{cases} $$ Now lets find the partial derivatives of the system for step 3.

Compute the partial derivatives

First, let's find \(f_1, f_2, f_3\) in terms of \(y_1, y_2,\) and \(y_3\) $$ f_1(t, y_1, y_2, y_3) = y_2 \\ f_2(t, y_1, y_2, y_3) = y_3 \\ f_3(t, y_1, y_2, y_3) = -ty_2 - 2y_1 $$ Now let's calculate the partial derivatives \(\frac{\partial f_i}{\partial y_j}\), for \(i,j=1,2,3\) $$ \frac{\partial f_1}{\partial y_1} = 0, \quad \frac{\partial f_1}{\partial y_2} = 1, \quad \frac{\partial f_1}{\partial y_3} = 0\\ \frac{\partial f_2}{\partial y_1} = 0, \quad \frac{\partial f_2}{\partial y_2} = 0, \quad \frac{\partial f_2}{\partial y_3} = 1\\ \frac{\partial f_3}{\partial y_1} = -2, \quad \frac{\partial f_3}{\partial y_2} = -t, \quad \frac{\partial f_3}{\partial y_3} = 0 $$ All the component functions and partial derivatives are continuous everywhere in the \((t, y_1, y_2, y_3)\) space, so Theorem 6.1's hypotheses hold for all points in the \((t, y_1, y_2, y_3)\) space. Finally, let's find the largest open rectangular region R according to Theorem 6.1.

Find the largest open rectangular region R

Since all component functions and partial derivatives are continuous everywhere in the space, the largest open rectangular region R is given by: $$ R = \{(t, y_1, y_2, y_3) \in \mathbb{R}^4 \} $$ This means that the theorem holds for all points in the \((t, y_1, y_2, y_3)\) space.

Key concepts

First-order ODEs

A first-order ordinary differential equation (ODE) is an equation that contains a first derivative, typically written as \( y' = f(t, y) \). It is first-order because the highest derivative appearing in the equation is of the first degree. These types of equations are crucial in modeling continuous processes ranging from physics to economics.
In the context of our problem, transforming a higher-order ODE into a system of first-order ODEs is a common technique to simplify and solve complex equations. By introducing new variables to represent each derivative, we effectively convert a single second-order ODE into multiple first-order ODEs.
This method allows us to leverage numerical methods and techniques that are specifically designed for first-order systems, making the analysis and solution simpler and more structured.

System of Differential Equations

A system of differential equations involves more than one differential equation that needs to be solved simultaneously. Such systems can describe the evolution of several interrelated phenomena over time. In practice, these variables interact, influencing each other's rates of change.
In the given exercise, we've transformed the original second-order equation into a system of three first-order ODEs. This is done by denoting \( y_1(t) \), \( y_2(t) \), and \( y_3(t) \) as functions representing \( y(t) \), its first derivative \( y'(t) \), and its second derivative \( y''(t) \) respectively. Each function in the system now describes part of the behavior of the original function and its derivatives.
By solving this system, we can determine the values of all functions at a particular point in time, allowing us to fully understand the dynamics of the original equation.

Partial Derivatives

Partial derivatives are used to study how a function changes as each of its input variables changes, while holding others constant. In a multivariable function, partial derivatives give us a sense of how the function varies with a specific input variable.
In our system of equations, we calculate partial derivatives for each function \( f_i \) with respect to each \( y_j \). These derivatives help us assess the continuity and smoothness of the functions in our system. Continuity is crucial because it ensures that the solutions to the differential equations behave well mathematically. In mathematical analysis, ensuring that functions and derivatives are continuous is often a prerequisite for applying important theorems such as theorem 6.1.
By confirming that all partial derivatives are continuous, we guarantee that the system is well-positioned for further analysis and solving.

Theorem 6.1

Theorem 6.1 concerns the conditions under which a solution to a system of differential equations exists and is unique. A central requirement of this theorem is the continuity of both the component functions and their partial derivatives.
In our exercise, we find that all components of our system's functions and their partial derivatives are continuous in the entire space of \((t, y_1, y_2, y_3)\). This indicates that there are no points where the conditions of Theorem 6.1 are violated. As such, the theorem assures us that there exists a unique solution over the entire space, which is often represented as an open rectangular region \(R\).
This is significant because in real-world applications, ensuring the existence and uniqueness of solutions is crucial in making reliable predictions and analyses based on our mathematical models.