Solving the Differential Equation dy/dx y/(1-x) x^2 - x using Advanced Techniques

Mathematics is a branch of advanced knowledge that often involves complex differential equations. One such equation is dy/dx y/(1-x) x^2 - x. This article will explore an advanced method to solve this equation: the use of an integrating factor.

Introduction

The differential equation in focus is dy/dx y/(1-x) x^2 - x. This is a first-order linear differential equation, which is common in various fields such as physics, engineering, and economics. Solving such equations using standard techniques can be challenging, and here, we will use an integrating factor to simplify the process.

Step-by-Step Solving Process

The general form of a first-order linear differential equation is dy/dx P(x)y Q(x). In the given equation, we have:

dy/dx P(x)y Q(x) where P(x) 1/(1-x) and Q(x) x^2 - x.

Step 1: Finding the Integrating Factor

First, we need to find the integrating factor (IF) which is given by the formula e^(∫P(x)dx). For this equation, we have:

e^(∫1/(1-x)dx) e^(-ln|1-x|) 1/(1-x).

Step 2: Multiplying the Equation by the Integrating Factor

Next, we multiply the entire differential equation by the integrating factor (IF) found in the previous step:

1/(1-x) * (dy/dx y/(1-x)) 1/(1-x) * (x^2 - x)

This step yields:

(1/(1-x))*dy/dx (1/(1-x))*y/(1-x) x^2 - x/(1-x)

The left side of the equation can be written as the derivative of the product of the integrating factor and y:

d/dx[(1/(1-x))*y] x^2 - x/(1-x)

Step 3: Integrating Both Sides

Now, integrate both sides with respect to x:

∫d/dx[(1/(1-x))*y] dx ∫(x^2 - x/(1-x)) dx

The left side simplifies to the product of the integrating factor and y:

(1/(1-x))*y ∫(x^2 - x/(1-x)) dx

Let's break down the right side integral:

Integral of x^2 with respect to x is x^3/3. Integral of x/(1-x) can be simplified using substitution. Let u 1-x, du -dx, x 1-u, and thus the integral becomes: ∫(1-u) du -u (1/2)u^2 -1 x (x-1)^2/2.

Combining these results:

(1/(1-x))*y x^3/3 - x (x-1)^2/2 C

Step 4: Solving for y

To find y, multiply both sides by 1/(1-x):

y (1/(1-x)) [x^3/3 - x (x-1)^2/2 C]

Conclusion

In conclusion, solving the differential equation dy/dx y/(1-x) x^2 - x using integrating factor technique involves three main steps: finding the integrating factor, multiplying the equation by the integrating factor, and integrating both sides. This method simplifies the process and provides a clear pathway to finding the general solution of the equation.

Keywords

Differential Equation Integrating Factor Advanced Mathematics

References

[1] Strang, G. (2010). Introduction to Linear Algebra. Wellesley-Cambridge Press.

[2] Boyce, W. E., DiPrima, R. C. (2017). Elementary Differential Equations and Boundary Value Problems. John Wiley Sons.