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? $$ t y^{\prime \prime}+\frac{1}{1+y+2 y^{\prime}}=e^{-t}, \quad y(2)=2, \quad y^{\prime}(2)=1 $$

Short answer

Question: Rewrite the given 2nd order scalar initial value problem as a system of first-order equations, compute the partial derivatives, and determine where in 3-dimensional (t, y1, y2)-space the hypotheses of Theorem 6.1 are not satisfied. Solution: After transforming the given equation into a system of first-order equations and computing the partial derivatives, the largest open rectangular region R where the hypotheses of Theorem 6.1 hold is R = {(t, y1, y2) | t ≠ 0}.

Step-by-step solution

Rewrite the given equation as a system of first-order equations

Define the new variables, \(y_1(t) = y(t)\) and \(y_2(t) = y'(t)\). Then, rewrite the given equation as \(y_1'(t) = y_2(t)\), \(y_2'(t) = \frac{e^{-t} - t^{-1}(1 + y_1(t) + 2y_2(t))}{t}\) The initial conditions become \(y_1(2) = 2\), \(y_2(2) = 1\) Now, the system of first-order equations is: \(\mathbf{y}'(t) = \left[\begin{array}{c}y_1'(t) \\ y_2'(t)\end{array}\right] = \left[\begin{array}{c}y_2(t) \\ \frac{e^{-t} - t^{-1}(1 + y_1(t) + 2y_2(t))}{t}\end{array}\right] = \mathbf{f}(t, \mathbf{y})\) with the initial condition \(\mathbf{y}(2) = \left[\begin{array}{c}2 \\ 1\end{array}\right]\).

Compute the \(n^2\) partial derivatives

We want to find the partial derivatives \(\frac{\partial f_i(t, y_1, y_2)}{\partial y_j}\) for \(i,j = 1, 2\). \(\frac{\partial f_1(t, y_1, y_2)}{\partial y_1} = 0\) \(\frac{\partial f_1(t, y_1, y_2)}{\partial y_2} = 1\) \(\frac{\partial f_2(t, y_1, y_2)}{\partial y_1} = \frac{-t^{-1}}{t} = -\frac{1}{t^2}\) \(\frac{\partial f_2(t, y_1, y_2)}{\partial y_2} = -\frac{2t^{-1}}{t} = -\frac{2}{t^2}\)

Determine where the hypotheses of Theorem 6.1 are not satisfied

Theorem 6.1 requires that all function values and partial derivatives are continuous in the region under consideration. Considering the partial derivatives we found in Step 2, we see that all the partial derivatives are continuous everywhere except at \(t=0\). Therefore, the largest open rectangular region \(R\) where the hypotheses of Theorem 6.1 hold is \(R = \{(t, y_1, y_2) \, | \, t \neq 0\}\).

Key concepts

Understanding Initial Value Problems

In the realm of differential equations, an initial value problem (IVP) is a fundamental concept. An IVP aims to find a function or a set of functions that not only satisfies a differential equation but also meets specified conditions at a given point, usually at the beginning, hence the term 'initial.' This extra information allows for a unique solution, tailoring the answer to a specific situation.

Consider a physical scenario as an analogy. If you tossed a ball into the air, the path it takes can be described by differential equations. However, to predict its exact trajectory, you need the initial position and velocity – these are your initial values. In the context of our exercise, we translate a higher-order differential equation into a system of first-order equations, which then allows us to apply methods and theorems specifically designed for first-order systems to solve the original problem.

When improving your understanding of IVPs, remember the importance of specifying both the initial condition and the equation it must satisfy. Addressing initial value problems effectively is crucial as it lays the groundwork for further understanding complex differential equations.

The Role of Partial Derivatives in Differential Equations

The concept of partial derivatives is central to multivariable calculus and appears frequently in the study of differential equations, especially when dealing with functions of several variables. A partial derivative represents the rate at which a function changes as one of its variables is varied, while all others are held constant.

To visualize, imagine adjusting the temperature on a hot day only concerning the time of day, ignoring other factors like cloud cover or humidity – this is akin to taking a partial derivative with respect to time. In our exercise, these derivatives are crucial for understanding the behavior of the components of the solution vector individually.

When practicing with partial derivatives, focus on being meticulous with your calculations, as they form the backbone for much of the analysis in these sorts of problems, such as stability and the behavior of solutions near critical points. Deepening your grasp on partial derivatives arms you with necessary tools to tackle more complex systems in your differential equations coursework.

Applying Theorem 6.1 Continuity to Differential Equations

Deciphering Theorem 6.1

Within the landscape of differential equations, Theorem 6.1 addresses the continuity requirements for the existence and uniqueness of solutions to an IVP. The theorem highlights that if the functions and their partial derivatives are continuous over a certain region, then a unique solution to the initial value problem exists within that region.

In the given problem, we analyze the system to determine where these conditions might falter. Specifically, the continuity of partial derivatives is compromised when the denominator, represented by time (\( t \)), approaches zero. Consequently, the theorem guides us to establish the largest domain wherein these critical functions retain their continuity.

Understanding Theorem 6.1’s implications, particularly the necessity for continuity of functions and their derivatives, is instrumental in discerning where solutions to differential equations may exist and behave predictably. Embrace the theorem as a test for examining the domains of solutions and as a prerequisite for safely deploying numerical methods for solving differential equations.