Solution of the differential equation `((dy)/(dx))-(y)/(x)=2x^(2)+3x+4` is

Options

A. `y=x^(3)+3x^(2)+4x log x +cx`
B. `y=x^(2)+3x +4 logx +c`
C. `y=x^(3)+3x^(2)+4 log x +c`
D. None

Answer

To solve the differential equation

\[ \frac{dy}{dx} - \frac{y}{x} = 2x^2 + 3x + 4, \]

we will follow the steps for solving a linear first-order differential equation.

### Step 1: Identify \( p(x) \) and \( q(x) \)

We can rewrite the equation in the standard form:

\[ \frac{dy}{dx} + \left(-\frac{1}{x}\right)y = 2x^2 + 3x + 4. \]

Here, we identify:
- \( p(x) = -\frac{1}{x} \)
- \( q(x) = 2x^2 + 3x + 4 \)

### Step 2: Find the Integrating Factor

The integrating factor \( \mu(x) \) is given by:

\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int -\frac{1}{x} \, dx} = e^{-\ln|x|} = \frac{1}{x}. \]

### Step 3: Multiply the Differential Equation by the Integrating Factor

Now, we multiply the entire differential equation by the integrating factor \( \frac{1}{x} \):

\[ \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} = \frac{2x^2 + 3x + 4}{x}. \]

This simplifies to:

\[ \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} = 2x + 3 + \frac{4}{x}. \]

### Step 4: Rewrite the Left Side

The left-hand side can be rewritten as:

\[ \frac{d}{dx}\left(\frac{y}{x}\right). \]

Thus, we have:

\[ \frac{d}{dx}\left(\frac{y}{x}\right) = 2x + 3 + \frac{4}{x}. \]

### Step 5: Integrate Both Sides

Now we integrate both sides with respect to \( x \):

\[ \int \frac{d}{dx}\left(\frac{y}{x}\right) \, dx = \int \left(2x + 3 + \frac{4}{x}\right) \, dx. \]

The left side integrates to:

\[ \frac{y}{x} = \int (2x + 3 + \frac{4}{x}) \, dx. \]

Now, we compute the right side:

\[ \int (2x) \, dx = x^2, \quad \int 3 \, dx = 3x, \quad \int \frac{4}{x} \, dx = 4 \ln|x|. \]

Combining these results gives:

\[ \frac{y}{x} = x^2 + 3x + 4 \ln|x| + C, \]

where \( C \) is the constant of integration.

### Step 6: Solve for \( y \)

To find \( y \), we multiply both sides by \( x \):

\[ y = x(x^2 + 3x + 4 \ln|x| + C) = x^3 + 3x^2 + 4x \ln|x| + Cx. \]

### Final Solution

Thus, the solution of the differential equation is:

\[ y = x^3 + 3x^2 + 4x \ln|x| + Cx. \]