Problem 39 - Advanced Engineering Mathematics

Question

Make up a differential equation that does not possess any real solutions.

Step-by-Step Solution

Answer

The differential equation \( \frac{dy}{dx} = \sqrt{-x^2 - y^2 - 1} \) has no real solutions since it results in complex numbers.

Step 1: Define the Form of the Differential Equation

First, let's consider a simple form of a first-order linear differential equation: \[\frac{dy}{dx} = f(x, y) \]To ensure that it has no real solutions, we'll choose a function \( f(x, y) \) that does not allow real-valued solutions. A good choice is to select a condition that leads to imaginary or complex numbers.

Step 2: Choosing a Function with No Real Solutions

To make sure the differential equation has no real solutions, one option is to create a scenario where the function under the square root is always negative, leading to an imaginary number. An example function is:\[\frac{dy}{dx} = \sqrt{-x^2 - y^2 - 1} \]This equation features a square root of terms that ensure non-real results.

Step 3: Explanation of the Chosen Function

In the equation \( \frac{dy}{dx} = \sqrt{-x^2 - y^2 - 1} \), the expression \(-x^2 - y^2 - 1\) under the square root is always negative for any real values of \(x\) and \(y\). Consequently, the result is a complex number since the square root of a negative number is not defined as a real number.

Step 4: Confirming No Real Solutions Exist

Given that the differential equation involves a square root of a negative number, which is complex, real solutions cannot exist. Therefore, no function \( y(x) \) satisfies this differential equation in the real number system.

Key Concepts

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part. The imaginary unit is denoted by the symbol \( i \), where \( i^2 = -1 \). A complex number is typically written as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
When dealing with differential equations, complex numbers arise in scenarios where solutions involve square roots of negative numbers.
For example, when the expression under the square root is negative, the solution to the equation may involve complex numbers instead of real ones. Complex numbers are not just a mathematical curiosity—they play a critical role in various fields, including engineering and physics. They help us solve equations and model phenomena that cannot be represented using only real numbers.

Real Solutions

Real solutions refer to those solutions of an equation or system that do not involve imaginary numbers. These solutions can be plotted on a number line. In the context of differential equations, real solutions provide the set of values that satisfy the equation such that the expression involved, such as a derivative or a function, remains real.
When the problem states to create a differential equation without real solutions, it implies that for every attempt to solve it using real numbers, the results lead to imaginary components.
This happens when the function in the differential equation results in values that make real number solutions impossible, necessitating the use of complex numbers to express these outcomes.

Imaginary Numbers

Imaginary numbers arise when dealing with the square roots of negative numbers. The basic imaginary unit is \( i \), which is defined as \( i = \sqrt{-1} \). An imaginary number can be expressed as \( bi \), where \( b \) is a real number and \( i \) is the imaginary unit.
Imaginary numbers themselves cannot be depicted on a typical number line because they do not satisfy the conditions of real numbers. Instead, they are visualized on an imaginary axis, perpendicular to the real axis, within the complex plane.
In many mathematical contexts, especially in differential equations, you encounter functions that lead to imaginary numbers when real solutions are sought impossible. This is common when the discriminant for a solution turns negative, hinting toward an imaginary resolution.

First-order Linear Differential Equations

First-order linear differential equations involve functions and their first derivatives. They have the general form \( \frac{dy}{dx} + P(x)y = Q(x) \), where \( P(x) \) and \( Q(x) \) are functions of \( x \).
Such equations are fundamental in understanding various dynamic and static systems and are often used in fields ranging from physics to economics.
The solution to these equations involves finding a function \( y(x) \) that satisfies this relationship.
First-order linear differential equations are usually solvable using methods like integrating factors or separable variables. However, when a non-traditional form ensures that real solutions do not exist, such as forming a situation leading to the square root of a negative number, these equations yield complex solutions. This was precisely the focus of the exercise, showing that under certain conditions, no real function \( y(x) \) can provide solutions for the differential equation.