Solution of Differential Equations Reducible to Bernoulli Form

Differential equations (DEs) are mathematical equations that relate functions with their derivatives. Among the various types of DEs, Bernoulli differential equations have a specific form that allows for a systematic solution approach. However, not all DEs are presented in the Bernoulli form initially. Some can be manipulated or reduced to a Bernoulli form, which then allows us to solve them using the Bernoulli equation method.

Bernoulli Differential Equation

A Bernoulli differential equation is a non-linear differential equation of the form:

$$ \frac{dy}{dx} + P(x)y = Q(x)y^n $$

where $n$ is any real number, and $P(x)$ and $Q(x)$ are continuous functions of $x$. When $n = 0$ or $n = 1$, the equation is linear and can be solved using linear methods. For other values of $n$, the equation is non-linear.

Reducing to Bernoulli Form

Some differential equations can be transformed into the Bernoulli form through substitution or algebraic manipulation. The goal is to identify a transformation that simplifies the equation into the standard Bernoulli form, which can then be solved using the method for Bernoulli equations.

Steps to Reduce a DE to Bernoulli Form:

1. Identify the non-linear term that prevents the equation from being in the Bernoulli form.
2. Perform an appropriate substitution or manipulation to eliminate the non-linear term.
3. Rewrite the equation in the standard Bernoulli form.
4. Solve the resulting Bernoulli equation.

Example

Consider the differential equation:

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

This equation is not in the Bernoulli form due to the $y^2$ term on the right-hand side. However, we can reduce it to Bernoulli form by making the substitution $v = y^{-1}$. This gives us:

$$ \frac{dv}{dx} = -y^{-2}\frac{dy}{dx} $$

Substituting $v$ and $\frac{dv}{dx}$ into the original equation, we get:

$$ -y^{-2}\frac{dy}{dx} - \frac{2}{x}y^{-1} = x^2 $$

Simplifying, we have:

$$ \frac{dv}{dx} + \frac{2}{x}v = x^2 $$

This is now in the standard Bernoulli form with $n = 0$, which is a linear first-order differential equation.

Solving the Bernoulli Equation

Once we have the DE in the Bernoulli form, we can solve it using the following steps:

1. Divide through by $y^n$ to get the equation in the form $\frac{dy}{dx} + P(x)y^{1-n} = Q(x)$.
2. Make the substitution $v = y^{1-n}$, which implies that $\frac{dv}{dx} = (1-n)y^{-n}\frac{dy}{dx}$.
3. Substitute $v$ and $\frac{dv}{dx}$ into the equation to get a linear DE in terms of $v$.
4. Solve the linear DE using an integrating factor or other methods for solving linear first-order DEs.
5. Back-substitute to find $y$ as a function of $x$.

Table of Differences and Important Points

Feature	General DE	Bernoulli DE	Reduced Bernoulli DE
Form	Can vary widely	$\frac{dy}{dx} + P(x)y = Q(x)y^n$	Initially not in Bernoulli form, but can be manipulated to match it
Linearity	Can be linear or non-linear	Non-linear (except for $n=0$ or $n=1$)	Non-linear term present, becomes linear after reduction
Solution Method	Depends on the form	Substitution $v = y^{1-n}$	Requires additional substitution or manipulation before applying Bernoulli method
Applicability	Broad	Specific to the form	Specific to DEs that can be manipulated into Bernoulli form

Conclusion

Understanding how to reduce a differential equation to Bernoulli form expands the range of DEs that can be solved using the systematic approach for Bernoulli equations. This technique is particularly useful when faced with non-linear terms that resemble the Bernoulli form. By mastering this method, students and professionals can tackle a wider array of problems in mathematics, physics, and engineering.