Solving Differential Equations: A Comprehensive Guide

Differential equations are fundamental in many fields including physics, engineering, and economics. Solving these equations involves finding functions that satisfy certain conditions. Understanding how to solve different types of differential equations can be crucial in applying mathematical models to real-world problems. This guide will explain how to solve a specific class of differential equations known as the Bernoulli equation. We will walk through the steps with an example and provide the necessary mathematical derivations.

Introduction to the Bernoulli Equation

The Bernoulli equation is a first-order nonlinear ordinary differential equation of the form:

[ y' P(x)y Q(x)y^n ]

where ( n ) is a real number, and ( P(x) ) and ( Q(x) ) are functions of ( x ). When ( n eq 0, 1 ), the equation is non-linear, making it more challenging to solve. However, by applying a substitution, it can be transformed into a linear differential equation.

Solving the Equation (frac{dy}{dx} a y - b y^2)

The given differential equation is:

[ frac{dy}{dx} a y - b y^2 ]

First, we need to rearrange the equation to match the form of a Bernoulli equation:

[ frac{dy}{dx} - a y -b y^2 ]

To solve this, we use a substitution ( u y^{-1} ). Then, we have:

[ u' -y^{-2} y' ]

Substituting into our equation, we get:

[ -u' - a u -b ]

This is a first-order linear differential equation. To solve it, we use the integrating factor method. The integrating factor is:

[ e^{int -a , dx} e^{-ax} ]

Multiplying both sides of the linear equation by the integrating factor ( e^{-ax} ), we obtain:

[ e^{-ax}(-u') - a e^{-ax} u -b e^{-ax} ]

Notice that the left-hand side is the derivative of ( u e^{-ax} ):

[ left( u e^{-ax} right)' -b e^{-ax} ]

Integrating both sides with respect to ( x ), we have:

[ u e^{-ax} b e^{-ax} C ]

Solving for ( u ), we get:

[ u b C e^{ax} ]

Recall that ( u y^{-1} ), so:

[ y^{-1} b C e^{ax} ]

Finally, solving for ( y ) gives:

[ y frac{1}{b C e^{ax}} ]

This is the general solution to the given differential equation.

Alternative Solution Using a Specific Form

For the convenience of analyzing the solution, we can rewrite ( C ) as ( pm e^C ) and drop the absolute value:

[ frac{y}{1 - by} A e^{ax} ]

Where ( A ) is a constant. Reciprocating this equation, we get:

[ frac{1}{y} - frac{b}{1 - by} frac{1}{A e^{ax}} ]

Rearranging and solving for ( y ), we obtain:

[ y frac{1}{b frac{1}{A} e^{-ax}} ]

Noting that ( A ) is just an arbitrary constant, we can denote it as ( B ), and write the final solution as:

[ y frac{1}{b B e^{-ax}} ]

This form highlights the role of the constant ( B ) more clearly.

Conclusion

The Bernoulli equation and solving it through substitution and integration are powerful analytical tools in mathematical physics and engineering. By understanding these methods, one can tackle more complex differential equations that arise in various practical applications.