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? $$ y^{\prime \prime}+e^{t} y=\ln |t|, \quad y(-1)=0, \quad y^{\prime}(-1)=-1 $$

Short answer

Question: Determine the largest open rectangular region R where the hypotheses of Theorem 6.1 hold for the given problem. Given problem: \(y''(t) + e^t y(t) = \ln|t|\), \(y(0) = 1\), and \(y'(0) = 0\). Answer: The largest open rectangular region R where the hypotheses of Theorem 6.1 hold is \((-\infty, 0) \times (-\infty, \infty) \times (-\infty, \infty) \cup (0, \infty) \times (-\infty, \infty) \times (-\infty, \infty)\), which is any region that does not include the y-axis (i.e., \(t=0\)).

Step-by-step solution

Converting the given initial value problem into a system of first-order equations

First, let's define \(y_1(t) = y(t)\), and \(y_2(t) = y'(t)\). That means \(\mathbf{y}(t) = \begin{bmatrix} y_1(t) \\ y_2(t) \end{bmatrix}\). Now we can write the given second-order equation as a system of first-order equations. Since \(y'(t) = y_2(t)\), the first equation is just \(y_1'(t) = y_2(t)\). Now, we have the second-order equation: \(y''(t) +e^t y(t) = \ln{|t|}\), which could be rewritten as \(y_2'(t) +e^t y_1(t) = \ln{|t|}\). Now the system of the first-order equations is: $$ y_1'(t) = y_2(t) \\ y_2'(t) + e^t y_1(t) = \ln\lvert{t}\rvert $$

Computing the required partial derivatives

To find the \(\mathbf{f}(t, \mathbf{y})\) vector we have that: $$ \mathbf{f}(t, \mathbf{y}) = \begin{bmatrix} y_2(t) \\ \ln\lvert{t}\rvert - e^t y_1(t) \end{bmatrix} $$ Now, let's compute the required partial derivatives: $$ \frac{\partial f_1(t, y_1, y_2)}{\partial y_1} = 0 \\ \frac{\partial f_1(t, y_1, y_2)}{\partial y_2} = 1 \\ \frac{\partial f_2(t, y_1, y_2)}{\partial y_1} = -e^t \\ \frac{\partial f_2(t, y_1, y_2)}{\partial y_2} = 0 $$

Finding points and region where hypotheses of the Theorem 6.1 are not satisfied

The component functions of \(\mathbf{f}(t, \mathbf{y})\) and the partial derivatives computed in step 2 are continuous for all \((t, y_1, y_2)\), except when \(t=0\). Thus, the hypotheses of Theorem 6.1 are not satisfied at points where \(t=0\). The largest open rectangular region R where the hypotheses of Theorem 6.1 hold would be any region that does not include the y-axis (i.e., \(t=0\)). For instance, R can be \((-\infty, 0) \times (-\infty, \infty) \times (-\infty, \infty) \cup (0, \infty) \times (-\infty, \infty) \times (-\infty, \infty)\).

Key concepts

First-order system of equations

In many mathematical problems, especially in differential equations, we often need to convert higher-order equations into a set of first-order equations. This makes the problems easier to handle and apply numerical methods if necessary. For example, given a second-order differential equation, we can define new variables that represent the function and its derivatives.
By creating a new vector of functions, we can transform the equation into a system of first-order differential equations.
This transformation involves organizing the original equations into a structure that only includes first derivatives. This step is fundamental in numerical analyses and solving systems using computational methods.

Partial derivatives

Partial derivatives are a critical concept in multivariable calculus, allowing us to understand how a function changes as each of its input variables changes, while keeping all other variables constant.
In the context of systems of differential equations, partial derivatives help us analyze the behavior of functions and their continuity. They are crucial in determining the local variation of component functions within our system, which can be first-order in nature.
These derivatives become particularly useful when applied to vector functions, as they allow us to explore the sensitivity of each function within the system to changes in specific variables.

Continuity

Continuity is a fundamental property for functions within a differential equation context, especially when evaluating the criteria for the existence and uniqueness of solutions.
In the given problem, continuity plays a vital role in ensuring that the partial derivatives and the function itself no longer abruptly change at certain points.
This is crucial because if a function isn't continuous, finding solutions becomes much more complex and sometimes even impossible in standard approaches. It's important that our function and its derivatives are continuous across the specified domains to meet the hypotheses of key theorems like Theorem 6.1 in the problem.

Rectangular region

A rectangular region in the context of differential equations is a specified subset of the domain where the functions and their derivatives behave well, that is, they are continuous and differentiable.
Such a region is often defined in terms of intervals for each variable, like \(a < t < b\), \(c < y_1 < d\), and encompasses our entire variable space without singularities or discontinuities.
For the largest open region, it is important that it includes those values where all the prerequisites for continuity and the differentiability hold true, ensuring the applicability of the existence and uniqueness theorems within these confines.