(x 
3
 −x) 
dx
dy
​
 −(3x 
2
 −1)y=x 
5
 −2x 
3
 +x solve separate variable differential equation 

We are given the differential equation:

\[
\left( x^3 - x \right) \frac{dx}{dy} - \left( 3x^2 - 1 \right) y = x^5 - 2x^3 + x
\]

To solve this, let's proceed step by step.

### Step 1: Rearrange the equation

We want to express the equation in a form where variables can be separated. Let's first isolate \(\frac{dx}{dy}\) on one side of the equation.

\[
\left( x^3 - x \right) \frac{dx}{dy} = (3x^2 - 1)y + (x^5 - 2x^3 + x)
\]

Now divide both sides of the equation by \((x^3 - x)\):

\[
\frac{dx}{dy} = \frac{(3x^2 - 1)y + (x^5 - 2x^3 + x)}{x^3 - x}
\]

### Step 2: Simplify the equation

Now let's focus on simplifying the right-hand side.

\[
\frac{dx}{dy} = \frac{(3x^2 - 1)y}{x^3 - x} + \frac{x^5 - 2x^3 + x}{x^3 - x}
\]

Notice that we can factor the denominator \(x^3 - x\) as:

\[
x^3 - x = x(x^2 - 1) = x(x - 1)(x + 1)
\]

We'll now separate terms and simplify them individually.

#### First term:

\[
\frac{(3x^2 - 1)y}{x(x - 1)(x + 1)}
\]

This cannot be easily separated as it contains both \(y\) and \(x\), so this method may not be suitable for direct separation of variables. Thus, a different approach may be needed (possibly an integrating factor or substitution).

Would you like me to continue with a different method, such as using an integrating factor or making a substitution to simplify the equation further?

---

### 5 Related Questions:
1. What is the method of separation of variables in differential equations?
2. How can integrating factors simplify first-order linear differential equations?
3. Can substitution methods like \(u = f(x)\) simplify differential equations with mixed terms?
4. What are homogeneous differential equations, and how are they solved?
5. How do you apply partial fractions to simplify complex rational expressions?

**Tip**: When solving differential equations, always check if terms can be factored or simplified before attempting to separate the variables.

(x^3 − x) dx/dy − (3x^2 − 1)y = x^5 − 2x^3 + x solve separate variable differential equation

Learn how to solve the differential equation (x^3 − x) dx/dy − (3x^2 − 1)y = x^5 − 2x^3 + x using separation of variables and integrating factors. This step-by-step guide simplifies complex algebraic and differential concepts, suitable for undergraduate-level students.

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