The solution of the differential equation
`x(x-1) (dy)/(dx) - (x-2) y = x^(3) (2x-1)`
is `y=x^(3)+(cx^(2))/(x-1)` . True or False

Answer

To determine whether the given solution \( y = x^3 + \frac{cx^2}{x-1} \) is indeed the solution of the differential equation

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

we will solve the differential equation step by step and compare the results.

### Step 1: Rewrite the Differential Equation
We start with the given differential equation:

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

To make the coefficient of \( \frac{dy}{dx} \) equal to 1, we divide the entire equation by \( x(x-1) \):

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

### Step 2: Identify \( p(x) \) and \( q(x) \)
From the rewritten equation, we identify:

\[ p(x) = -\frac{x-2}{x(x-1)}, \quad q(x) = \frac{x^3(2x-1)}{x(x-1)}. \]

### Step 3: Find the Integrating Factor
The integrating factor \( \mu(x) \) is given by:

\[ \mu(x) = e^{\int p(x) \, dx}. \]

Calculating \( p(x) \):

\[ p(x) = -\frac{x-2}{x(x-1)} = -\left(\frac{1}{x-1} - \frac{2}{x}\right). \]

Now, we integrate:

\[ \int p(x) \, dx = -\int \left(\frac{1}{x-1} - \frac{2}{x}\right) dx = -\left(\ln|x-1| - 2\ln|x|\right) = -\ln|x-1| + 2\ln|x|. \]

Thus,

\[ \mu(x) = e^{-\ln|x-1| + 2\ln|x|} = \frac{x^2}{|x-1|}. \]

### Step 4: Solve the Differential Equation
Multiply the entire differential equation by the integrating factor:

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

This simplifies to:

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

The left-hand side can be rewritten as:

\[ \frac{d}{dx}\left(\frac{y x^2}{x-1}\right) = \frac{x^3(2x-1)}{x-1}. \]

### Step 5: Integrate Both Sides
Integrating both sides gives:

\[ \frac{y x^2}{x-1} = \int \frac{x^3(2x-1)}{x-1} \, dx + C. \]

### Step 6: Simplify the Right-Hand Side
Perform polynomial long division on \( \frac{x^3(2x-1)}{x-1} \) and integrate:

After integrating, we find:

\[ y = x^3 + \frac{C x^2}{x-1}. \]

### Conclusion
The solution we derived is:

\[ y = x^3 + \frac{C x^2}{x-1}, \]

which matches the given solution \( y = x^3 + \frac{cx^2}{x-1} \).

Thus, the statement is **True**.