Problem 39 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. The system of differential equations \(d x / d t=-x+x y^{2}\) and \(d y / d t=y-x^{2} y\) requires initial conditions for both \(x(0)\) and \(y(0)\) to determine a unique solution.

Step-by-Step Solution

Answer

True, both initial conditions are needed for a unique solution according to the theorem.

Step 1: Understand the Problem

We are given two differential equations: \(\frac{dx}{dt} = -x + xy^2\) and \(\frac{dy}{dt} = y - x^2y\). We need to determine if initial conditions for both \(x(0)\) and \(y(0)\) are necessary to find a unique solution.

Step 2: Analyze the System

This is a system of first-order ordinary differential equations with two variables, \(x\) and \(y\). According to the existence and uniqueness theorem for systems of differential equations (Picard-Lindelöf theorem), a unique solution exists if the functions and their partial derivatives are continuous, and initial conditions are provided for all variables.

Step 3: Verify the Conditions

The right-hand sides of both equations, \(-x + xy^2\) and \(y - x^2y\), are polynomials, hence continuous everywhere. Additionally, we would need partial derivatives of these functions to be continuous, which they are, since they are also polynomials. Thus, all conditions for the theorem are satisfied.

Step 4: Apply the Existence and Uniqueness Theorem

Since the functions and their partial derivatives are continuous, providing initial conditions \(x(0)\) and \(y(0)\) ensures that a unique solution exists according to the existence and uniqueness theorem.

Key Concepts

Initial Conditions

In the context of differential equations, initial conditions are values given for a function and its derivatives at a specific point. These conditions are crucial to uniquely solving a differential equation.
Without initial conditions, there are infinitely many solutions to a differential equation.

• Initial conditions specify the state of the function at the beginning of the period of interest.
• They determine the particular solution that corresponds to the specific scenario described by the problem.

For example, in the system given, initial conditions for both functions, such as \(x(0) = a\) and \(y(0) = b\), allow us to pinpoint the exact curve on the solution plane, leading to a unique solution.Therefore, defining initial conditions means you have adequately described the starting state, which is essential for arriving at a single valid solution for the differential equations in question.

Existence and Uniqueness Theorem

The Existence and Uniqueness Theorem is a fundamental concept in solving differential equations. This theorem guarantees that a differential equation has a unique solution if certain conditions are met.
These conditions often include continuity and specific initial conditions.

• This theorem is important because it gives you confidence that your solution not only exists but is also the only one.
• For a single or a system of differential equations, it ensures that the problem is well-posed.

The theorem requires the function involved in the differential equation (and, for systems, their partial derivatives) to be continuous over a specific range. In our case, the functions \(-x + xy^2\) and \(y - x^2y\) are polynomials, therefore continuous everywhere.
This means the prerequisites for the existence and uniqueness are fulfilled, confirming that with provided initial conditions, the solutions are consistent and unique.

Picard-Lindelöf Theorem

The Picard-Lindelöf Theorem is a specific application of the Existence and Uniqueness Theorem suitable for initial value problems. This theorem offers a method for finding unique solutions to ordinary differential equations with initial conditions.

• It asserts that if both the function and its derivative are continuous in a neighborhood of the initial value, a unique solution exists in some interval around this point.
• This also applies to systems like the one in the original exercise.

The theorem is very practical, as it provides steps for constructing the solution using an iterative method known as Picard iteration.
As noted, the equations provided meet the conditions of the Picard-Lindelöf Theorem, since the equations and their partial derivatives are continuous which ensures that for any initial conditions \(x(0)\) and \(y(0)\), a unique trajectory or solution path can be determined.
In summary, this theorem provides both the foundation and verification method for solving the initial value problems uniquely and effectively.