## Solving the Differential Equation: (x^3 - x)dy/dx - (3x^2 - 1)y = x^5 - 2x^3 + x

This article will guide you through the process of solving the given first-order linear differential equation.

### Understanding the Equation

The equation is a **first-order linear differential equation** because it involves the first derivative of y (dy/dx) and both y and its derivative appear linearly. The equation is of the form:

**P(x) dy/dx + Q(x)y = R(x)**

where:

- P(x) = x^3 - x
- Q(x) = -(3x^2 - 1)
- R(x) = x^5 - 2x^3 + x

### Solving the Equation

We will use the method of **integrating factors** to solve this equation.

**1. Finding the Integrating Factor:**

The integrating factor is given by:

**μ(x) = exp(∫(Q(x)/P(x)) dx)**

In this case:

μ(x) = exp(∫(-(3x^2 - 1)/(x^3 - x)) dx)

We can simplify the integrand by factoring the denominator:

μ(x) = exp(∫(-(3x^2 - 1)/(x(x^2 - 1))) dx) = exp(∫(-(3x^2 - 1)/(x(x - 1)(x + 1))) dx)

Now we can use partial fractions to break down the integrand:

(-(3x^2 - 1)/(x(x - 1)(x + 1))) = (A/x) + (B/(x - 1)) + (C/(x + 1))

Solving for A, B, and C, we get:

A = 1, B = -2, C = 1

Therefore:

μ(x) = exp(∫(1/x - 2/(x - 1) + 1/(x + 1)) dx)

Integrating:

μ(x) = exp(ln|x| - 2ln|x - 1| + ln|x + 1|) = exp(ln(|x(x + 1)/(x - 1)^2|)) = **|x(x + 1)/(x - 1)^2|**

We can choose the positive sign for simplicity, since the integrating factor is used for multiplication.

**2. Multiplying the Equation by the Integrating Factor:**

Multiplying the original equation by the integrating factor:

[x(x + 1)/(x - 1)^2] [(x^3 - x)dy/dx - (3x^2 - 1)y] = [x(x + 1)/(x - 1)^2] [x^5 - 2x^3 + x]

**3. Simplifying the Equation:**

Notice that the left-hand side now becomes the derivative of a product:

d/dx [y(x(x + 1)/(x - 1)^2)] = [x(x + 1)/(x - 1)^2] [x^5 - 2x^3 + x]

**4. Integrating Both Sides:**

Integrating both sides with respect to x:

∫ d/dx [y(x(x + 1)/(x - 1)^2)] dx = ∫ [x(x + 1)/(x - 1)^2] [x^5 - 2x^3 + x] dx

This simplifies to:

y(x(x + 1)/(x - 1)^2) = ∫ [x(x + 1)/(x - 1)^2] [x^5 - 2x^3 + x] dx + C

**5. Solving for y:**

Finally, solve for y:

**y = [(x - 1)^2/(x(x + 1))] ∫ [x(x + 1)/(x - 1)^2] [x^5 - 2x^3 + x] dx + C[(x - 1)^2/(x(x + 1))]**

### Conclusion

The solution to the differential equation is given by the above expression, where C is an arbitrary constant of integration. The integral on the right-hand side needs to be evaluated to get the explicit solution. This process demonstrates the power of the integrating factor method for solving linear first-order differential equations.