Question

State whether each of the following problems is an initial-value or a boundary-value problem: (a) \(\quad y^{\prime \prime}+3 y^{\prime}+2 y=f(x)\), \(y(0)=1, y^{\prime}(0)=2 .\) (b) \(y^{\prime \prime}+(\sin x) y=0\), \(y(0)=1, y(1)=0 .\) (c) \(y^{\prime \prime}+4 y=\cos x\), \(y(1)=2, y^{\prime}(1)=0 .\) (d) \(y^{\prime \prime}-2 y^{\prime}+3 y=e^{x}\), \(y(0)=1, y^{\prime}(1)=2\)

Short answer

Question: Determine whether each of the following problems is an initial-value problem or a boundary-value problem. a) \(y''+3y'+2y=f(x)\), where \(y(0)=1\) and \(y'(0)=2\) b) \(y''+(\sin x)y=0\), where \(y(0)=1\) and \(y(1)=0\) c) \(y''+4y=\cos x\), where \(y(1)=2\) and \(y'(1)=0\) d) \(y''-2y'+3y=e^x\), where \(y(0)=1\) and \(y'(1)=2\) Answer: a) Initial-Value Problem b) Boundary-Value Problem c) Initial-Value Problem d) Boundary-Value Problem

Step-by-step solution

Problem (a)

The given problem is \(y''+3y'+2y=f(x)\), where \(y(0)=1\) and \(y'(0)=2\). Since both conditions are given at the same point \(x=0\), this is an initial-value problem.

Problem (b)

The given problem is \(y''+(\sin x)y=0\), where \(y(0)=1\) and \(y(1)=0\). Since the conditions are given at different points in the domain (at \(x=0\) and \(x=1\)), this is a boundary-value problem.

Problem (c)

The given problem is \(y''+4y=\cos x\), where \(y(1)=2\) and \(y'(1)=0\). Since both conditions are given at the same point \(x=1\), this is an initial-value problem.

Problem (d)

The given problem is \(y''-2y'+3y=e^x\), where \(y(0)=1\) and \(y'(1)=2\). Since the conditions are given at different points in the domain (at \(x=0\) and \(x=1\)), this is a boundary-value problem.

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are fundamental in the realms of mathematics and sciences, as they are used to describe the behavior of various physical, biological, and economic systems. At their core, ODEs are equations that involve derivatives of a function with respect to one independent variable. They come in many orders, where the order is determined by the highest derivative present in the equation.

For example, an equation like \( y''+3y'+2y=f(x) \) involves the second derivative of the function \( y \) with respect to \( x \) and is therefore called a second-order ODE. Solving such an equation typically requires the application of specific methods tailored to the type of ODE and the nature of the function \( f(x) \).

When approaching ODEs, we aim to find a function \( y(x) \) that satisfies the given differential equation through all the specified conditions. Understanding the general and particular solutions to these equations is crucial for comprehensively grasping the concepts behind many physical phenomena and processes.

Initial Conditions

Initial conditions are an essential aspect of solving ordinary differential equations. They provide specific values of the function or its derivatives at a particular point, which helps in finding the unique solution to an ODE. These conditions are a fundamental part of initial-value problems (IVPs), where the goal is to solve the ODE with the given starting parameters.

For instance, consider the ODE from our exercise \( y''+3y'+2y=f(x) \), accompanied by the conditions \( y(0)=1 \) and \( y'(0)=2 \). These statements describe the value of the function \( y \) and its first derivative at the point \( x=0 \). Such information allows the solver to pin down the exact behavior of the solution near \( x=0 \) and hence throughout its domain, assuming the existence and uniqueness of the solution are guaranteed. Without initial conditions, most ordinary differential equations would have infinitely many solutions; with them, we typically can identify the one solution that fits the given scenario.

Boundary Conditions

Boundary conditions, in contrast to initial conditions, are used in problems where the value of the solution or its derivatives is known at different points within a particular domain. These are crucial in the context of boundary-value problems (BVPs), which often arise in the study of physical systems with constraints at multiple points.

An example within the scope of our exercise is the second ODE \( y''+(\sin x)y=0 \), with \( y(0)=1 \) and \( y(1)=0 \). Here, the values of \( y \) are specified at both ends of the interval from 0 to 1. Such conditions do not specify the immediate behavior after the initial point but rather impose requirements that must be satisfied at the edges or boundaries of the interval. Solving boundary-value problems can be more complex than initial-value problems due to the nature of the requirements spread across the domain. The application of boundary conditions plays a key role in disciplines such as engineering, physics, and mathematics, where phenomena are often studied within a finite region or under specific constraints.