Question

Apply Theorem 2.3 .1 to the initial value problem $$ y^{\prime}+p(x) y=q(x), \quad y\left(x_{0}\right)=y_{0} $$ for a linear equation, and compare the conclusions that can be drawn from it to those that follow from Theorem 2.1.2.

Short answer

Question: Apply Theorem 2.3.1 to the following initial value problem (IVP) for a linear ordinary differential equation (ODE) and compare the conclusions that can be drawn to those that follow from Theorem 2.1.2: $$ y'+p(x)y=q(x), \quad y(x_0)=y_0 $$ Answer: Both Theorem 2.3.1 and Theorem 2.1.2 ensure the existence and uniqueness of a solution for the given IVP, provided that the required conditions of continuity hold. Theorem 2.3.1 is specialized for linear ODEs, whereas Theorem 2.1.2 can be applied to any first-order ODE satisfying the continuity conditions.

Step-by-step solution

Identify the given IVP

We have the linear equation: $$ y'+p(x)y=q(x), \quad y(x_0)=y_0 $$

Verify the conditions of Theorem 2.3.1

Suppose \(p(x)\) and \(q(x)\) are continuous functions on some interval containing \(x_0\). Then, by Theorem 2.3.1, there exists a unique solution \(y(x)\) to the IVP on that interval.

Compare the conclusions of Theorem 2.3.1 with Theorem 2.1.2

First, let's rewrite the given ODE in the form \(y'=f(x, y)\): $$ y' = -p(x)y + q(x) $$ Here, \(f(x, y) = -p(x)y + q(x)\). Since both \(p(x)\) and \(q(x)\) are continuous, then \(f(x, y)\) is continuous as well. Now, we need to determine \(\frac{\partial f}{\partial y}\): $$ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (-p(x)y + q(x)) = -p(x) $$ Since \(p(x)\) is continuous, \(\frac{\partial f}{\partial y}\) itself is continuous. By Theorem 2.1.2, since \(f(x, y)\) and \(\frac{\partial f}{\partial y}\) are continuous in a rectangle \(R\) containing \((x_0, y_0)\), there exists a unique solution of the IVP on an interval \(I\) containing \(x_0\). The conclusions from both Theorem 2.3.1 and Theorem 2.1.2 are the same: they both ensure the existence and uniqueness of a solution for the IVP. The difference is that Theorem 2.3.1 is specialized for linear ODEs, whereas Theorem 2.1.2 can be applied to any first-order ODE that satisfies the required conditions of continuity.

Key concepts

Theorem 2.3.1

Theorem 2.3.1 is a vital tool when dealing with linear initial value problems (IVPs). This theorem specifically addresses linear differential equations of the form:

• \( y' + p(x)y = q(x) \)

where \( y(x_0) = y_0 \) is the initial condition. The importance of Theorem 2.3.1 lies in its ability to establish both the existence and uniqueness of solutions.

For the theorem to apply, the functions \( p(x) \) and \( q(x) \) must be continuous over some interval that includes the initial point, \( x_0 \). When these conditions are satisfied, one can reliably conclude that a unique solution exists within this interval.

Theorem 2.3.1 thereby ensures that the problem is well-posed and can be solved with confidence.

Linear Differential Equations

Linear differential equations form an important category within differential equations and are often easier to manage due to their structural properties. The general linear differential equation is:

• \( y' + p(x)y = q(x) \)

This type of equation is called 'linear' because the dependent variable \( y \) and all its derivatives appear to the power of one.

In many physical and engineering problems, linear differential equations adequately describe the behavior of systems under study. Their linearity allows for the superposition principle, which states that the sum of two solutions is also a solution. This is a powerful property that significantly simplifies both the solution process and the interpretation of results.

Understanding the characteristics of linear differential equations is crucial for building foundational knowledge in calculus and differential equations.

Existence and Uniqueness Theorem

The Existence and Uniqueness Theorem provides the backbone for solving many ordinary differential equations (ODEs). It states that, under specific conditions, an initial value problem has one and only one solution.

For an equation of the form \( y' = f(x, y) \), if:

• \( f(x, y) \) is continuous, and
• the partial derivative \( \frac{\partial f}{\partial y} \) is continuous

in some rectangle surrounding the point \( (x_0, y_0) \), then the IVP has a unique solution in a neighborhood about \( x_0 \).

This theorem reassures that the solution not only exists but is also unique, preventing the occurrence of multiple solutions that could otherwise occur due to the arbitrary choices often involved in integration.

First-order ODEs

First-order ordinary differential equations (ODEs) are the simplest form of differential equations, involving a first derivative but no higher derivatives. They appear in the form:

• \( y' = f(x, y) \)

These equations are foundational and arise in various practical problems, such as exponential growth, cooling laws, and motion dynamics.

The general strategy for solving a first-order ODE is transforming it into a separable equation, applying integration, and using any given initial values to solve for constants of integration.

With a solid understanding of first-order ODEs, one can explore more complex forms, develop analytical skills, and apply these concepts to real-world scenarios.