Solution of Differential Equations by Equations Reducible to the Variable Separable Method

Differential equations (DEs) are equations that involve derivatives of unknown functions. They are fundamental in describing various phenomena in engineering, physics, economics, and other sciences. One of the simplest methods to solve first-order ordinary differential equations (ODEs) is the variable separable method. However, not all DEs are presented in a separable form. Some can be transformed or reduced to a separable form through algebraic manipulation or substitution. This process is known as making an equation reducible to the variable separable method.

Understanding Variable Separable Method

A first-order ODE is said to be separable if it can be written in the form:

$$ \frac{dy}{dx} = g(x)h(y) $$

The solution involves separating the variables x and y on different sides of the equation:

$$ \frac{1}{h(y)}dy = g(x)dx $$

Integrating both sides gives the general solution:

$$ \int \frac{1}{h(y)}dy = \int g(x)dx + C $$

where C is the constant of integration.

Making Equations Reducible to Variable Separable Form

Some DEs may not initially appear to be separable, but with appropriate manipulation, they can be reduced to a separable form. This can be done through:

1. Algebraic Manipulation: Rearranging terms or factoring to separate variables.
2. Substitution: Introducing a new variable to transform the DE into a separable form.

Table of Techniques to Reduce DEs to Separable Form

Technique	Description	Example
Algebraic Manipulation	Rearrange or factor the DE to separate variables.	$\frac{dy}{dx} = \frac{x^2 + y^2}{xy}$ can be written as $\frac{dy}{dx} = \frac{x}{y} + \frac{y}{x}$
Substitution	Introduce a new variable to simplify the DE.	For $\frac{dy}{dx} = \frac{x + y}{x - y}$, use $v = \frac{y}{x}$

Formulas and Examples

Example 1: Algebraic Manipulation

Consider the DE:

$$ \frac{dy}{dx} = \frac{x^2 + y^2}{xy} $$

This can be manipulated to:

$$ \frac{dy}{dx} = \frac{x}{y} + \frac{y}{x} $$

Now, separate the variables:

$$ \frac{y}{x}dy = \left(1 + \frac{y^2}{x^2}\right)dx $$

Integrate both sides:

$$ \int \frac{y}{x}dy = \int dx + \int \frac{y^2}{x^2}dx $$

Example 2: Substitution

Consider the DE:

$$ \frac{dy}{dx} = \frac{x + y}{x - y} $$

Let's make the substitution $v = \frac{y}{x}$, then $y = vx$ and $\frac{dy}{dx} = v + x\frac{dv}{dx}$.

Substitute into the original DE:

$$ v + x\frac{dv}{dx} = \frac{1 + v}{1 - v} $$

Now, separate the variables:

$$ (1 - v^2)dv = \frac{1}{x}dx $$

Integrate both sides:

$$ \int (1 - v^2)dv = \int \frac{1}{x}dx $$

Important Points to Remember

• Not all DEs are separable, but some can be made separable through manipulation or substitution.
• Always check for the possibility of separating variables before attempting more complex methods.
• Substitutions should simplify the DE and make it easier to separate and integrate.
• After solving the DE with the substitution, remember to revert back to the original variables to find the general solution.

By understanding these techniques and practicing various examples, one can become proficient in solving first-order ODEs that are reducible to the variable separable method, which is a valuable skill in mathematics and its applications.