Solving a Differential Equation Using Bernoulli's Equation: A Step-by-Step Guide

When faced with a differential equation that can be transformed into a more manageable form, such as a Bernoulli's equation, the process can be both enlightening and rewarding. This article will guide you step-by-step through the solution of the differential equation:

The Problem at Hand

The given differential equation is:

dy2xmiddotdx x e-x2 y3dx

To solve this equation, we will start by rearranging the terms and applying the Bernoulli's equation method.

Rearrangement

First, we rearrange the given equation:

Rearranged Equation

y2xmiddotdx x e-x2 y3dx.

We can separate the terms involving to one side and the terms involving to the other:

y2xmiddotdx / 3 x e-x2 dx.

This simplifies to:

y-2xmiddotdx x e-x2 dx.

Substitution

Next, we use the substitution method. Let:

u -2

Then, we take the derivative of u with respect to and use it in the equation:

u-1y -2y-3y -2 dy/dx -2 y-3x e-x2 dx.

Multiplying both sides by -2:

u 4xu 2xe-x2.

First-Order Linear Differential Equation

The resulting equation is a first-order linear differential equation:

u' 4xu 2xe-x2.

To solve this equation, we utilize the integrating factor method. The integrating factor, , can be found by solving:

eint 4xdx e -2x2.

Using the Integrating Factor

Multiply the entire differential equation by the integrating factor, e-2x2:

e-2x2u 4xe-2x2u 2xe-x2 - 2x2.

The left side of the equation is now the derivative of the product u with respect to :

d/dx(e-2x2u) 2xe-x2 - 2x2.

Integrate both sides of the equation:

e-2x2u int 2xe-x2 - 2x2 dx.

Using a substitution method, let v -x2, then dv -2dx, and the integral becomes:

e-2x2u int -e2v dv -e2v -e-2x2 C, where C is the constant of integration.

Finally, we solve for u:

u -1 Ce2x2.

Back Substitution

Recalling that u y-2, we perform the back substitution:

y-2 -1 Ce2x2.

This equation can be rewritten to solve for y:

y plusmn;1/#x221A;(-1 Ce2x2).

This is the solution to the original differential equation, provided that the denominator is non-zero.

The Bottom Line

By transforming the given equation into a Bernoulli's equation, we were able to use the integrating factor method to arrive at a solution. This process not only offers insight into the nature of the equation but also showcases the power of substitution and integration techniques in solving complex differential equations.

Related Keywords

Bernoulli's Equation, Differential Equation, Integration Factor