Solving a Differential Equation Through Recurrence Formula and Integrating Factor

In this article, we explore the solution of the differential equation y' - xy^2y 0. This process involves the use of a recurrence formula and the application of an integrating factor, demonstrating the elegance and complexity in solving such nonlinear differential equations. Let's break down the steps involved and provide a detailed explanation.

Introduction to the Problem

Consider the differential equation y' - xy^2y 0. Our goal is to find the solution for y as a function of x. We will use a combination of recurrence formula and the integrating factor method to tackle this problem.

Step 1: Initial Assumptions and Simplifications

Let's start by assuming a specific form for the solution, namely, y_1 Ax^2Bx C. We will then differentiate y_1 with respect to x to find y_1' and substitute both y_1 and y_1' into the given differential equation.

Starting with y_1 2AxB, we differentiate to get:

y_1' 2A - 2Ax^2 - Bx(2Ax B)

Substituting into the differential equation:

2A - 2Ax^2 - Bx(2Ax B) - x(2AxB) - 2Ax^2 - Bx(2Ax B) 0

Simplifying further, we obtain:

2A - 2Ax^2 - Bx(2Ax B) - 2Ax^2 - Bx(2Ax B) - 2Ax^2 - Bx(2Ax B) 0

After simplification, we find that:

Bx(2Ax B) 0

This implies that:

B 0 and 2A(2Ax B) - 2Ax^2 - Bx(2Ax B) 0

From which we further deduce:

A -C

Therefore, our first solution becomes:

y_1 -Cx^2 - C C(x^2 - 1)

Step 2: Using the Integrating Factor Method

Now, let's consider y_1 C(x^2 - 1) as a known solution and introduce a new function y y_1z. Then we have:

y' (2xz z') (x^2 - 1) z

Substitute into the original differential equation to find:

2xz(x^2 - 1)z - x(2xz(x^2 - 1)z) - 2(x^2 - 1)z 0

After simplification, we get:

2xz(x^2 - 1)z - 2x^2z(x^2 - 1)z - 2(x^2 - 1)z 0

Further simplification leads to:

z(x^2 - 1)(4x - (x^2 - 1)x) 0

This implies:

z(x^2 - 1) - xz(x^2 - 5) 0

To solve this, we can use an integrating factor p(z). Let's set:

p(z) (x^2 - 1)^2e^{-x^2/2}

Now, we solve the differential equation for p(z):

p'(z) -x^3 - 5x (x^2 - 1) p(z)

Differential equation for p(z) becomes:

d(p(z)/(x^2 - 1)^2 e^{-x^2/2}) 0

Integrating both sides, we get:

p(z) C_1 (x^2 - 1)^2 e^{x^2/2}

Thus, we find:

z C_1 e^{-x^2/2} / (x^2 - 1)^2

Therefore, the general solution for y is:

y (x^2 - 1)z C_1 (x^2 - 1) e^{-x^2/2} / (x^2 - 1)^2

Which simplifies to:

y C_1 e^{-x^2/2} / (x^2 - 1)

Conclusion

In this article, we have successfully solved the differential equation y' - xy^2y 0 using a combination of a recurrence formula and the integrating factor method. The solution, y C_1 e^{-x^2/2} / (x^2 - 1), provides insight into the behavior of the function and its dependence on the variable x.

For further reading, consider exploring nonlinear differential equations and techniques for solving them, as well as the application of integrating factors in more complex scenarios.