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? $$ y^{\prime \prime \prime}+y^{\prime}+y^{2}=0, \quad y(-1)=0, \quad y^{\prime}(-1)=1, \quad y^{\prime \prime}(-1)=0 $$

Short answer

Question: Determine the largest open rectangular region R where the hypotheses of Theorem 6.1 hold for the given initial value problem. The largest open rectangular region R that contains the initial point (-1,0,1,0) and where the hypotheses of Theorem 6.1 hold is given by R = {(t, y_1, y_2, y_3) | a < t < b, y_1 ∈ ℝ, y_2 ∈ ℝ, y_3 ∈ ℝ} for any a < -1 < b.

Step-by-step solution

Rewrite the problem as a system of equations

First, let \(y_1(t) = y(t)\), \(y_2(t) = y'(t)\), and \(y_3(t) = y''(t)\). Then this initial problem can be rewritten as a system of first-order equations, using the following substitutions: \begin{align*} y_1'(t) &= y_2(t) \\ y_2'(t) &= y_3(t) \\ y_3'(t) &= - y_2(t) - y_1^2(t) \\ \end{align*} Now let \(\mathbf{y}(t) = \begin{bmatrix} y_1(t) \\ y_2(t) \\ y_3(t) \end{bmatrix}\) and \(\mathbf{f}(t, \mathbf{y}) = \begin{bmatrix} y_2(t) \\ y_3(t) \\ -y_2(t) - y_1^2(t) \end{bmatrix}\). The system of equations can be rewritten as: \(\mathbf{y}'(t) = \mathbf{f}(t,\mathbf{y})\) with initial conditions \(\mathbf{y}(-1) = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\).

Compute the partial derivatives

We can now compute the required partial derivatives as follows: \begin{align*} \frac{\partial f_1(t, y_1, y_2, y_3)}{\partial y_1} &= 0 \\ \frac{\partial f_1(t, y_1, y_2, y_3)}{\partial y_2} &= 1 \\ \frac{\partial f_1(t, y_1, y_2, y_3)}{\partial y_3} &= 0 \\ \frac{\partial f_2(t, y_1, y_2, y_3)}{\partial y_1} &= 0 \\ \frac{\partial f_2(t, y_1, y_2, y_3)}{\partial y_2} &= 0 \\ \frac{\partial f_2(t, y_1, y_2, y_3)}{\partial y_3} &= 1 \\ \frac{\partial f_3(t, y_1, y_2, y_3)}{\partial y_1} &= -2y_1 \\ \frac{\partial f_3(t, y_1, y_2, y_3)}{\partial y_2} &= -1 \\ \frac{\partial f_3(t, y_1, y_2, y_3)}{\partial y_3} &= 0 \\ \end{align*}

Determine the largest open rectangular region R

According to Theorem 6.1, we need to find where the components of \(\mathbf{f}(t, \mathbf{y})\) and its partial derivatives are continuous. In this case, all the components of \(\mathbf{f}(t, \mathbf{y})\) and all partial derivatives are continuous functions except \(\partial f_3\left(t, y_{1}, y_{2}, y_{3}\right) / \partial y_{1} = -2y_1\), which depends on \(y_1\). This means that the region \(R\) should not be affected by \(t\), \(y_2\), or \(y_3\). The largest open rectangular region \(R\) that contains the initial point \((-1,0,1,0)\) and where the hypotheses of Theorem 6.1 hold is \(R = \{ (t, y_1, y_2, y_3) \mid a < t < b, y_1 \in \mathbb{R}, y_2 \in \mathbb{R}, y_3 \in \mathbb{R} \}\) for any \(a < -1 < b\).

Key concepts

System of Differential Equations

When we encounter a higher order differential equation, such as a third-order equation like in our exercise, one effective method to tackle it is to convert it into a system of first-order differential equations. This is precisely what occurs when we set \( y_1(t) = y(t) \) and associate successive derivatives with subsequent functions, such as \( y_2(t) = y^{\prime}(t) \) and \( y_3(t) = y^{\prime\prime}(t) \).

This approach breaks down the complex relationships into a set of simpler equations that are interrelated, where the derivative of each function within the vector \( \mathbf{y} \) is expressed as a function of \( t \) and the other functions within \( \mathbf{y} \). It's akin to untangling a complicated knot by methodically addressing each entwined loop one by one. An advantage of working with this system is that it allows for the application of powerful techniques and theorems tailored to systems of equations, which can reveal insights about the behavior of solutions that might not be clear from the original equation.

Partial Derivatives

Partial derivatives play a crucial role in understanding the behavior of a system of differential equations. In our case, we are taking the derivative of a function with respect to one variable while keeping the other variables constant. Imagine painting a multicolored landscape; partial derivatives would be akin to focusing on how changing the shade of just one color (variable) impacts the whole scene (function), while all other colors remain unchanged.

The computed partial derivatives tell us how the respective functions in the system, \( \mathbf{f}(t, \mathbf{y}) \), change when one of the variables \( y_i \) is varied slightly. In technical terms, they represent the Jacobian matrix of the system, and their importance cannot be overstated—they are involved in the analysis of system stability, convergence of numerical methods, and in the formulation of conditions for the existence and uniqueness of solutions to differential equations.

Theorem Application

The application of theorems in differential equations is akin to having a master key while exploring a vast mansion; theorems open the doors to rooms filled with the theoretical understanding necessary to navigate complex mathematical spaces.

For example, Theorem 6.1, which is alluded to in the exercise, likely refers to a result that provides conditions under which a unique solution exists for a system of differential equations. Conditions of this theorem typically involve the continuity and boundedness of the function \( \mathbf{f} \) and its partial derivatives. As demonstrated in the solution, we examine the continuity of partial derivatives \( \partial f_i / \partial y_j \) to establish where the theorem applies. By identifying the 'largest open rectangular region' where the theorem's conditions are satisfied, we can assure that in this zone, the system behaves predictably, and solutions can be trusted to not suddenly veer off into mathematical chaos.

Continuity of Functions

The concept of continuity is quite intuitive—imagine drawing a curve without lifting the pencil from the paper. In mathematical terms, a function is continuous if small changes in the input result in small changes in the output, without any abrupt jumps or breaks.

In the context of differential equations, the continuity of functions is vital because it underpins the theorems that guarantee the existence of solutions and their properties. When a function, such as \( f_i(t, y_1, ..., y_n) \), or its partial derivatives, are not continuous, it's similar to hitting a pothole on a smooth road; the journey (solution) can be disrupted. In the solution we developed, the continuity of all functions and the partial derivatives \( \partial f_i / \partial y_j \) was confirmed, except for \( \partial f_3 / \partial y_1 = -2y_1 \), which is a linear function of \( y_1 \) and inherently continuous. Therefore, our system doesn't encounter any 'mathematical potholes' in the region \( R \) regarding continuity, making it feasible to apply Theorem 6.1.