Solution of the differential equation `(dy)/(dx)+(1)/(x)y=3x` is

Options

A. x
B. log x
C. 0
D. `oo`

Answer

To solve the differential equation

\[ \frac{dy}{dx} + \frac{1}{x}y = 3x, \]

we will follow the method of integrating factors. Here are the steps:

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

In the given equation, we can identify:

- \( p = \frac{1}{x} \)
- \( q = 3x \)

### Step 2: Calculate the Integrating Factor

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

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

Calculating the integral:

\[ \int \frac{1}{x} \, dx = \log |x|. \]

Thus, the integrating factor becomes:

\[ \mu(x) = e^{\log |x|} = |x| \quad \text{(we can take } x \text{ since we assume } x > 0\text{)}. \]

So, we have:

\[ \mu(x) = x. \]

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

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

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

### Step 4: Rewrite the Left Side as a Derivative

The left-hand side can be rewritten as the derivative of a product:

\[ \frac{d}{dx}(xy) = 3x^2. \]

### Step 5: Integrate Both Sides

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

\[ \int \frac{d}{dx}(xy) \, dx = \int 3x^2 \, dx. \]

This gives us:

\[ xy = x^3 + C, \]

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

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

Now, we solve for \( y \):

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

### Final Solution

Thus, the solution of the differential equation is:

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

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