Question

Discuss the existence and uniqueness of the solution of the following initial- value problems \(\begin{array}{ll}\text { (a) } & y_{1}^{\prime}=y_{1}+e^{t} y_{2}, \\ & y_{2}^{\prime}=(\sin t) y_{1}+t^{2} y_{2}, \\ & y_{1}(0)=1, \quad y_{2}(0)=0 . \\ \text { (b) } & y_{1}^{\prime}=y_{1}+t y_{2}+e^{t} y_{3}, \\\ y_{2}^{\prime}=y_{2}-t^{2} y_{3}, & \\ y_{3}^{\prime}=t y_{1}-y_{2}+y_{3} . & \\\ y_{1}(0)=1, y_{2}(0)=0, y_{3}(0)=0 .\end{array}\)

Short answer

Answer: Yes, both initial-value problems (a) and (b) have a unique solution. This is because they satisfy the conditions of the Picard-Lindelöf theorem, as the matrix functions and their partial derivatives with respect to y are continuous in the entire ty-plane.

Step-by-step solution

The given initial-value problem can be written in matrix form as: \(\begin{pmatrix} y_{1}^{\prime} \\ y_{2}^{\prime} \end{pmatrix} = \begin{pmatrix} 1 & e^t \\ \sin t & t^2 \end{pmatrix} \cdot \begin{pmatrix} y_{1} \\ y_{2} \end{pmatrix}\), with initial conditions \(y_{1}(0) = 1\) and \(y_{2}(0) = 0\). #a) Step 2: Verify the continuity of F and its partial derivative#

The matrix function F(t, y) = \(\begin{pmatrix} 1 & e^t \\ \sin t & t^2 \end{pmatrix}\) is continuous in the entire ty-plane. Now, we have to find the partial derivative of F(t,y) with respect to y and check its continuity: \(\frac{\partial F}{\partial y} = \begin{pmatrix} \frac{\partial}{\partial y_{1}}(1) & \frac{\partial}{\partial y_{1}}(e^t) \\ \frac{\partial}{\partial y_{2}}(\sin t) & \frac{\partial}{\partial y_{2}}(t^2) \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\). This matrix function is also continuous in the entire ty-plane. #a) Step 3: Apply Picard-Lindelöf theorem#

As both F(t, y) and its partial derivative with respect to y are continuous in the ty-plane, it satisifies the conditions of Picard-Lindelöf theorem. Thus, there exists a unique solution for the initial-value problem (a) in some interval around t0 = 0. #b) Step 1: Write the given system in matrix form#

The given initial-value problem can be written in matrix form as: \(\begin{pmatrix} y_{1}^{\prime} \\ y_{2}^{\prime} \\ y_{3}^{\prime} \end{pmatrix} = \begin{pmatrix} 1 & t & e^t \\ 0 & 1 & -t^2 \\ t & -1 & 1 \end{pmatrix} \cdot \begin{pmatrix} y_{1} \\ y_{2} \\ y_{3} \end{pmatrix}\), with initial conditions \(y_{1}(0) = 1\), \(y_{2}(0) = 0\), and \(y_{3}(0) = 0\). #b) Step 2: Verify the continuity of F and its partial derivative#

The matrix function F(t, y) = \(\begin{pmatrix} 1 & t & e^t \\ 0 & 1 & -t^2 \\ t & -1 & 1 \end{pmatrix}\) is continuous in the entire ty-plane. Now, we have to find the partial derivative of F(t,y) with respect to y and check its continuity: \(\frac{\partial F}{\partial y} = \begin{pmatrix} \frac{\partial}{\partial y_{1}}(1) & \frac{\partial}{\partial y_{1}}(t) & \frac{\partial}{\partial y_{1}}(e^t) \\ \frac{\partial}{\partial y_{2}}(0) & \frac{\partial}{\partial y_{2}}(1) & \frac{\partial}{\partial y_{2}}(-t^2) \\ \frac{\partial}{\partial y_{3}}(t) & \frac{\partial}{\partial y_{3}}(-1) & \frac{\partial}{\partial y_{3}}(1) \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\). This matrix function is also continuous in the entire ty-plane. #b) Step 3: Apply Picard-Lindelöf theorem#

As both F(t, y) and its partial derivative with respect to y are continuous in the ty-plane, it satisifies the conditions of Picard-Lindelöf theorem. Thus, there exists a unique solution for the initial-value problem (b) in some interval around t0 = 0.

Key concepts

Existence and Uniqueness of Solutions

Understanding the existence and uniqueness of solutions to differential equations is essential in mathematics. The Picard-Lindelöf theorem is a cornerstone in this field, providing necessary conditions under which a solution to an initial value problem (IVP) exists and is unique.
When approaching an IVP, it's crucial to check the continuity of the function and its partial derivatives. In the case of our exercise, matrix functions for both (a) and (b) scenarios showed continuous behavior in the domain - a clear indication that uniqueness and existence could be assured. In simpler terms, if you have an IVP and can prove these conditions, the theorem gives you a thumbs-up that there's exactly one solution path you can follow, starting from your given initial point.

Initial Value Problems

In the realm of differential equations, an Initial Value Problem (IVP) refers to a scenario where we know the state of the system at a specific starting time. For instance, in our textbook problems, we're given the initial conditions for the functions y_1, y_2, and y_3 at time t=0. Think of it as being handed a snapshot of a dynamic process at its birth and being tasked to forecast its future. The IVP consists of a differential equation along with the initial states, which play a pivotal role in predicting the behavior of the system as time progresses.
For both exercises (a) and (b), the statement of the IVP set the stage for us to unleash the power of the Picard-Lindelöf theorem.

Ordinary Differential Equations

The study of Ordinary Differential Equations (ODEs) is about understanding how quantities change with respect to one variable, usually time. They're not just a bunch of mysterious symbols – these equations are the language of change, allowing us to describe the motion of planets, the growth of populations, or the decay of radioactive materials, among countless other processes in science and engineering.
The examples provided in the exercise are ODEs that represent a system where the rate of change of each function is interconnected with others. Solving such equations sometimes feels like unlocking a cryptic message that nature has left for us to decipher!

Matrix Formulation of Differential Equations

The matrix formulation of differential equations is a game-changer when you're dealing with multiple interrelated functions. It's like organizing a team where each member's effort depends on the others – instead of looking at them individually, you harness the power of matrices to capture the whole picture.
In our exercise, writing the system of ODEs in matrix form streamlined the process, enabling us to evaluate properties like continuity more effectively. With each function's derivative on one side and a matrix multiplying the function vector on the other, analyzing and solving complex interactions becomes a structured and efficient task.

Continuity Conditions

The concept of continuity conditions is not just confined to making sure there are no breaks in a line on a graph. In the context of ODEs and the Picard-Lindelöf Theorem, it serves as a critical checkpoint. We must ensure that the functions we deal with don't have any abrupt changes or discontinuities within the region of interest.
For our initial value problems in the matrix formulation, we verified whether the function F(t, y) and its partial derivatives were continuous across the entire plane. This step isn't just a formality – it ensures that our solutions won't suddenly hit a mathematical pothole. By ensuring continuity, we create a smooth roadway for our solutions, free from unpredictable jumps or holes.