(y Ln X)^-1 Dy/dx=(x/y+1)^2

4 min read Jun 17, 2024

Solving the Differential Equation: (y ln x)^-1 dy/dx = (x/y + 1)^2

This article explores the solution process for the given differential equation:

(y ln x)^-1 dy/dx = (x/y + 1)^2

This equation is classified as a first-order non-linear differential equation, making its solution more intricate compared to simpler linear equations. We will employ a combination of techniques to solve it:

1. Rearranging the Equation

First, we aim to rewrite the equation in a more manageable form. We can start by simplifying the left-hand side and multiplying both sides by y ln x:

dy/dx = (y ln x)(x/y + 1)^2

2. Substitution Method

To make the equation more amenable to integration, we introduce a substitution. Let:

u = y ln x

Taking the derivative of both sides with respect to x, we get:

du/dx = (dy/dx) ln x + y/x

Now, we can express dy/dx in terms of u and its derivative:

dy/dx = (du/dx - y/x) / ln x

Substituting this back into our original equation:

(du/dx - y/x) / ln x = u (x/y + 1)^2

3. Simplifying the Equation

We can further simplify the equation by substituting y = u / ln x:

(du/dx - u/(x ln x)) / ln x = u (x(ln x)/u + 1)^2

This simplifies to:

du/dx - u/(x ln x) = u (x ln x + u)^2 / ln x

4. Separating Variables

Now, we can rearrange the equation to separate the variables u and x:

du / (u(x ln x + u)^2 / ln x + u/(x ln x)) = dx

This can be rewritten as:

du / (u(x ln x + u)^2 + u) = dx / (x ln x)

5. Integrating Both Sides

Now, we can integrate both sides of the equation. The integration of the right-hand side is straightforward:

∫ dx / (x ln x) = ln(ln x) + C1

The left-hand side requires partial fraction decomposition. The result will involve a combination of logarithmic and arctangent functions. After integration, we will have an equation relating u and x.

6. Back Substitution

Finally, we need to substitute back our original variable y: