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Chapter 7: Problem 24

a. Write down a differential equation that has no solution. b. Write down a differential equation that has exactly one solution.

Short Answer

Expert verified
For (a): \( y' = y^2, \) \( y(0) = -1;\) no solution. For (b): \( y' = -2y, \) \( y(0) = 3;\) one solution.

Step by step solution

01

Understanding a Differential Equation with No Solution

To write a differential equation with no solution, consider an inherently inconsistent equation. A common example in differential equations is trying to specify an impossible initial condition for a given type of differential equation.
02

Example of a Differential Equation with No Solution

An example of such an equation is the initial value problem: \( y' = y^2, \) with the initial condition \( y(0) = -1. \) Since \( y^2 \) is always non-negative, \( y' = y^2 \) implies a non-negative velocity; therefore, it cannot lead to a negative solution.
03

Understanding a Differential Equation with Exactly One Solution

For a differential equation to have a unique solution, it typically needs to satisfy certain conditions like the Lipschitz condition in a given interval. This ensures the well-posedness and uniqueness of the solution.
04

Example of a Differential Equation with Exactly One Solution

Consider the initial value problem: \( y' = -2y, \) with the initial condition \( y(0) = 3. \) This is a linear first-order differential equation. According to the theory of linear differential equations, such an equation with a specified initial condition has exactly one solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Initial Value Problem
An initial value problem in differential equations is a problem where we seek a solution to a differential equation that satisfies a specific condition at the start, known as the initial condition. To understand it better:
  • It consists of a differential equation itself, like \( y' = y^2 \).
  • Additionally, it has an initial condition, such as \( y(0) = -1 \).
The initial condition is crucial because it determines where the solution curve starts on the coordinate graph. Engineering and physics problems often involve initial value problems, where conditions at time zero (or time \( t_0 \)) are known. Such problems provide a unique path or trajectory for the solution by specifying it at one exact point.
Unique Solution
Having a unique solution means that exactly one function satisfies both the differential equation and the initial condition within a particular interval. Not all differential equations with initial conditions yield unique solutions. We can identify some key aspects:
  • The differential equation needs to be well-posed. This means that, in addition to having a solution, it should be sensibly solvable under small variations in initial conditions.
  • The initial value problem \( y' = -2y \) with \( y(0) = 3 \) is a typical case where there's a unique solution. Its uniqueness is assured by satisfying the conditions required for the existence and uniqueness theorem.
This solution can be understood as a single path that cannot be duplicated or altered regardless of analysis methods within the bounds of the given conditions.
Lipschitz Condition
The Lipschitz condition is a mathematical criterion used to guarantee the uniqueness of solutions to differential equations. It states that a function must not change too rapidly, or in more technical terms, differ by more than a constant proportional to the change in its inputs. Here are some insights:
  • If a function \( f(x, y) \) satisfies the Lipschitz condition, it will generally ensure the uniqueness of the solution to the differential equation \( y' = f(x, y) \).
  • This condition can be checked by finding if there exists a constant \( L \) such that for any two points \( y_1 \) and \( y_2 \), \( |f(x, y_1) - f(x, y_2)| \leq L |y_1 - y_2| \).
  • For example, in the differential equation \( y' = -2y \), the function is linear and satisfies the Lipschitz condition, hence ensures one unique solution for the initial condition given.
Thus, the Lipschitz condition helps mathematicians and scientists predict whether initial value problems will behave predictably with a singular unique solution.

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Most popular questions from this chapter

A falling skydiver experiences a "drag force" due to air resistance. The drag force increases with the magnitude of the velocity \(v\) of the diver and is approximately equal to \(k v^{2}\), where \(k\) is a constant. As a result, \(v\) is given approximately by $$ v(t)=-\sqrt{\frac{m g}{k}} \tanh \left(\sqrt{\frac{k g}{m}} t\right) $$ where \(m\) is the mass of the skydiver and \(g\) is the acceleration due to gravity. Find \(\lim _{t \rightarrow \infty} v(t) .\) The speed of the skydiver can never exceed the magnitude of this

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Find the general solution of the separable differential equation. $$ \frac{2 y}{y^{2}+1} \frac{d y}{d x}=\frac{1}{x^{2}} $$

Plot the graph of \(y=\sin ^{-1}(\sin x)\) with windows \(-6 \leq x \leq\) 6 and \(-1.6 \leq y \leq 1.6 .\) Explain the appearance of the graph.
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