Question

For each initial value problem, determine the largest \(t\)-interval on which Theorem \(3.1\) guarantees the existence of a unique solution. $$y^{\prime \prime}+y^{\prime}+3 t y=\tan t, \quad y(\pi)=1, \quad y^{\prime}(\pi)=-1$$

Short answer

The largest t-interval on which the existence and uniqueness of the solution to the given initial value problem is guaranteed by Theorem 3.1 is (0, 2π).

Step-by-step solution

Recall the statement of Theorem 3.1

Theorem 3.1 (Existence and Uniqueness for Second Order Linear ODEs): Suppose that \(\frac{\partial p}{\partial t}, \frac{\partial q}{\partial t},\) and \(\frac{\partial r}{\partial t}\) are continuous on a rectangle \(R = \{(t, y) : t_0 \le t \le t_0 + a, -\infty < y < \infty\}\). Then, there exists a unique solution \(y = y(t)\) of the initial value problem $$y'' + p(t)y' + q(t)y = r(t), \quad y(t_0) = y_0, \quad y'(t_0) = y_1,$$ for each set of initial conditions \(y_0\) and \(y_1\). Now, our task is to find the largest t-interval where the existence and uniqueness of the solution for the given IVP is guaranteed by Theorem 3.1.

Write the given IVP in standard form

The given IVP can be written in the standard form for a second-order linear inhomogeneous differential equation as: $$y'' + 1\cdot y'+ 3ty = \tan t, \quad y(\pi) = 1, \quad y'(\pi) = -1$$ Comparing with the general form, we have \(p(t) = 1\), \(q(t) = 3t\), and \(r(t) = \tan t\).

Check for continuity of functions

To apply Theorem 3.1, we need to ensure that \(p(t)\), \(q(t)\), and \(r(t)\) are continuous on a rectangle R. We know that \(p(t)\) and \(q(t)\) are continuous for all t, but we need to pay attention to the function \(r(t) = \tan t\). The tangent function has vertical asymptotes where the cosine function is zero, which occurs at \(t = \left(n+\frac{1}{2}\right)\pi\) for any integer \(n\). Thus, the tangent function is continuous on intervals between consecutive asymptotes, i.e., intervals of the form \(\left(n\pi, (n+1)\pi\right)\).

Determine the largest t-interval for existence and uniqueness of the solution

Given our initial condition \(y(\pi) = 1\), we must look for the largest interval surrounding \(t = \pi\) on which the tangent function is continuous. The interval in question is \(\left(0,\, 2\pi\right)\), which does not include the end points since tangent function has its vertical asymptote at \(t=2\pi\). So, the largest t-interval on which Theorem 3.1 guarantees the existence and uniqueness of the solution is \((0,\, 2\pi)\).

Key concepts

Existence and Uniqueness Theorem

The Existence and Uniqueness Theorem is a fundamental concept in the study of differential equations. This theorem provides conditions under which a differential equation will have a unique solution passing through a given point in its domain. For second-order linear differential equations, the theorem guarantees that if the coefficients of the equation and the forcing function are continuous, a unique solution exists on the specified domain.
For our particular problem, we have a differential equation of the form:

• \(y'' + p(t)y' + q(t)y = r(t), \quad y(t_0) = y_0, \quad y'(t_0) = y_1\).

The theorem uses a concept of a rectangle in the \((t, y)\) plane. The solution exists and is unique within this region as long as the functions involved, such as \(p(t)\), \(q(t)\), and \(r(t)\), are continuous there. The primary aim is to identify the largest interval around a point where these functions do not have any discontinuities.
This theorem is extremely useful because it assures us that under certain conditions, we can predict the behavior of solutions even without explicitly solving the differential equation.

Initial Value Problems

Initial Value Problems (IVPs) are a specific type of problem where the solution to a differential equation is sought under certain initial conditions. In the context of second-order linear differential equations, these initial conditions usually involve setting both the function and its first derivative to specific values at a point \(t_0\).
Let's consider the given IVP:

• \(y'' + y' + 3ty = \tan t, \quad y(\pi) = 1, \quad y'(\pi) = -1\).

The initial conditions are \(y(\pi) = 1\) and \(y'(\pi) = -1\), indicating the value of the function and its rate of change at \(t = \pi\).
Specifying such conditions allows us to pin down a specific solution from potentially many solutions of the differential equation. The existence and uniqueness theorem ensures that only one function satisfies both the differential equation and these initial conditions on a suitable interval.
This process of "anchoring" a solution with initial values makes IVPs a reliable method for modelling real-world processes where starting conditions are known.

Continuity

Continuity is a critical factor when solving differential equations, particularly when applying the Existence and Uniqueness Theorem. A function is continuous at a point if it is defined there and if its value approaches the function's value as we near that point. When dealing with an interval, we require continuity throughout that domain.
In our problem, the functions \(p(t) = 1\) and \(q(t) = 3t\) are continuous over all real numbers, read as they don't have any breaks or holes.

• However, \(r(t) = \tan t\) introduces a challenge. It includes discontinuities at \(t = \left(n+\frac{1}{2}\right)\pi\) for integers \(n\), where it experiences vertical asymptotes.

These points occur because \(\tan t = \frac{\sin t}{\cos t}\), and \(\cos t\) equates to zero at these points. Therefore, to ensure continuity for \(r(t)\), we must avoid these asymptotes.
Identifying the intervals where these discontinuities do not appear allows us to apply the Existence and Uniqueness Theorem effectively. Continuity, hence, helps in determining that the largest possible interval for our problem is \((0, 2\pi)\), not including the endpoints.

Differential Equations Solutions

Solving differential equations can be daunting, but the process becomes simpler with a structured approach. Differential equations are mathematical equations that involve one or more functions and their derivatives. The goal is to find a function \(y(t)\) that satisfies both the differential equation and any given conditions.
In our exercise, we are tasked to consider:

• \(y'' + y' + 3ty = \tan t, \quad y(\pi) = 1, \quad y'(\pi) = -1\).

This is an inhomogeneous differential equation due to the presence of a non-zero term \(\tan t\). Determining a solution involves confirming that the functions \(p(t), q(t), r(t)\) meet the criteria set by relevant theorems. Once verified, the solution is unique within that interval.

• Using techniques such as variation of parameters or undetermined coefficients can help find particular solutions to inhomogeneous equations.

In addition to understanding the mathematical aspects, consider what the solution represents physically or practically. The modelling power of differential equations lies in their ability to translate real-world phenomena into a solvable framework, providing critical insights into the dynamics of systems.