Solving the Differential Equation x2 x2 - 1 dy/dx x x2 - 1 y x2 - 1

In this comprehensive guide, we will explore the method of solving the differential equation:

x2 x2 - 1 dy/dx   x x2 - 1 y  x2 - 1

Let's proceed step-by-step to arrive at the final solution.

Step 1: Standard Form

First, we rewrite the equation in a more standard form:

(frac{dy}{dx} - frac{x x2 - 1}{x2 x2 - 1} y  frac{x2 - 1}{x2 x2 - 1})

This simplifies to:

(frac{dy}{dx} - frac{1}{x2} y  1)

This is a first-order linear ordinary differential equation of the form:

(frac{dy}{dx}   P(x) y  Q(x))

where:

(P(x) frac{x x2 - 1}{x2 x2 - 1} frac{x^3 - x}{x^4 - x^2} frac{1}{x} - frac{1}{x^2 - 1})

and

(Q(x) frac{1}{x2})

Step 2: Integrating Factor

To solve this, we first find the integrating factor (mu(x)):

(mu(x) e^{int P(x) , dx})

Calculating (P(x)):

(P(x) frac{x^3 - x}{x^4 - x^2} frac{1}{x} - frac{1}{x^2 - 1})

Integrating (P(x)):

(int P(x) , dx int left(frac{1}{x} - frac{1}{x^2 - 1}right) , dx)

The integral of (frac{1}{x}) is (ln |x|), and for (frac{1}{x^2 - 1}) we can use partial fractions:

(frac{1}{x^2 - 1} frac{1/2}{x - 1} - frac{1/2}{x 1})

Thus, we have:

(int P(x) , dx ln |x| - frac{1}{2} ln |x - 1| - frac{1}{2} ln |x 1| C)

Combining the logarithmic terms, we find:

(int P(x) , dx ln left| x sqrt{frac{x - 1}{x 1}} right| C)

So the integrating factor is:

(mu(x) e^{ln left| x sqrt{frac{x - 1}{x 1}} right|} x sqrt{frac{x - 1}{x 1}})

Step 3: Multiply by the Integrating Factor

Multiplying the entire differential equation by the integrating factor:

(x sqrt{frac{x - 1}{x 1}} cdot frac{dy}{dx} - x sqrt{frac{x - 1}{x 1}} cdot frac{1}{x^2} y x sqrt{frac{x - 1}{x 1}})

This can be rewritten using the product rule:

(frac{d}{dx} left( y cdot x sqrt{frac{x - 1}{x 1}} right) x sqrt{frac{x - 1}{x 1}})

Step 4: Integrate Both Sides

Integrating both sides with respect to (x):

(y cdot x sqrt{frac{x - 1}{x 1}} int x sqrt{frac{x - 1}{x 1}} , dx C)

To solve for (y), we need to evaluate the integral:

(y frac{1}{x sqrt{frac{x - 1}{x 1}}} left( int x sqrt{frac{x - 1}{x 1}} , dx C right))

The final solution will depend on the evaluation of the integral and simplification based on the initial conditions if provided.

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