Formation of Differential Equations (DE)

Differential equations (DEs) are mathematical equations that relate functions to their derivatives. They are fundamental in describing various phenomena in engineering, physics, economics, and other sciences. The formation of differential equations involves expressing a relationship involving derivatives of an unknown function.

Basic Concepts

Before diving into the formation of DEs, let's review some basic concepts:

• Differential Equation: An equation involving derivatives of one or more functions.
• Order: The highest derivative present in the DE.
• Degree: The highest power of the highest order derivative, provided the DE is polynomial in derivatives.
• Solution: A function that satisfies the DE when substituted into it.

Formation from a Given Relation

To form a DE from a given relation involving one or more dependent variables and their derivatives, follow these steps:

1. Identify all the independent and dependent variables in the relation.
2. Differentiate the relation with respect to the independent variable(s) as many times as necessary to eliminate arbitrary constants.
3. Express the relation in the form of a DE by isolating one of the derivatives.

Example

Consider the family of curves given by the equation:

$$ y = Ae^{3x} + Be^{-2x} $$

where (A) and (B) are arbitrary constants. To form a DE, we need to eliminate (A) and (B).

1. Differentiate the equation with respect to (x):

$$ \frac{dy}{dx} = 3Ae^{3x} - 2Be^{-2x} $$

1. Differentiate again:

$$ \frac{d^2y}{dx^2} = 9Ae^{3x} + 4Be^{-2x} $$

1. Now we have three equations and two unknowns ((A) and (B)). We can solve for (A) and (B) in terms of (y) and its derivatives:

From the first differentiation:

$$ A = \frac{1}{3}e^{-3x}\frac{dy}{dx} + \frac{2}{3}Be^{-5x} $$

$$ B = \frac{1}{2}e^{2x}\frac{dy}{dx} - \frac{3}{2}Ae^{5x} $$

1. Substitute (A) and (B) into the second derivative equation to eliminate them:

After some algebraic manipulation, we get:

$$ \frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = 0 $$

This is the second-order linear homogeneous DE corresponding to the given family of curves.

Table of Differences and Important Points

Feature	Ordinary DE	Partial DE
Definition	Involves derivatives with respect to one independent variable.	Involves derivatives with respect to two or more independent variables.
Example	$\frac{dy}{dx} = 3y$	$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
Order	Determined by the highest derivative with respect to the single independent variable.	Determined by the highest derivative with respect to any of the independent variables.
Degree	Power of the highest order derivative when the DE is polynomial in derivatives.	Power of the highest order derivative when the DE is polynomial in derivatives.
Solution	A function of one variable.	A function of multiple variables.

Formulas

The general form of an (n)-th order ordinary differential equation is:

$$ F\left(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \ldots, \frac{d^ny}{dx^n}\right) = 0 $$

For partial differential equations, the general form is:

$$ F\left(x_1, x_2, \ldots, x_n, u, \frac{\partial u}{\partial x_1}, \frac{\partial u}{\partial x_2}, \ldots, \frac{\partial^2 u}{\partial x_1 \partial x_2}, \ldots\right) = 0 $$

Conclusion

The formation of differential equations is a systematic process of eliminating arbitrary constants from a given relation by differentiation. The resulting DE captures the essence of the relationship between the variables and their rates of change. Understanding how to form and solve DEs is crucial for modeling and analyzing dynamic systems across various scientific disciplines.