If `y_(1)(x)` is a solution of the differential equation `(dy)/(dx)-f(x)y = 0`, then a solution of the differential equation `(dy)/(dx) + f(x) y = r(x)` is

Options

A. `y = (1)/(y_(1)(x))int r(x) y_(1)(x) dx +(c)/(y_(1)(x))`
B. `y = y_(1)(x) int (r(x))/(y_(1)(x))dx + c`
C. `y = int r(x) y_(1)(x) dx + c`
D. None of these

Explanation

To solve the problem, we need to find a solution for the differential equation

\[ \frac{dy}{dx} + f(x) y = r(x) \]

given that \( y_1(x) \) is a solution of the differential equation

\[ \frac{dy}{dx} - f(x) y = 0. \]

### Step-by-Step Solution:

1. **Identify the first differential equation**:
The first equation is

\[ \frac{dy}{dx} - f(x) y = 0. \]

Rearranging gives:

\[ \frac{dy}{dx} = f(x) y. \]

2. **Separate variables**:
We can separate the variables:

\[ \frac{1}{y} dy = f(x) dx. \]

3. **Integrate both sides**:
Integrating both sides yields:

\[ \int \frac{1}{y} dy = \int f(x) dx. \]

This gives:

\[ \log |y| = \int f(x) dx + C_1, \]

where \( C_1 \) is the constant of integration.

4. **Exponentiate to solve for \( y \)**:
Exponentiating both sides results in:

\[ y = e^{\int f(x) dx + C_1} = C e^{\int f(x) dx}, \]

where \( C = e^{C_1} \) is a constant.

Since \( y_1(x) \) is a solution, we can write:

\[ y_1(x) = C e^{\int f(x) dx}. \]

5. **Now, consider the second differential equation**:
The second equation is

\[ \frac{dy}{dx} + f(x) y = r(x). \]

This is a linear first-order differential equation.

6. **Find the integrating factor**:
The integrating factor \( \mu(x) \) is given by:

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

7. **Multiply through by the integrating factor**:
We multiply the entire equation by \( \mu(x) \):

\[ e^{\int f(x) dx} \frac{dy}{dx} + e^{\int f(x) dx} f(x) y = e^{\int f(x) dx} r(x). \]

The left-hand side can be rewritten as:

\[ \frac{d}{dx} \left( y e^{\int f(x) dx} \right) = e^{\int f(x) dx} r(x). \]

8. **Integrate both sides**:
Integrating both sides gives:

\[ y e^{\int f(x) dx} = \int e^{\int f(x) dx} r(x) dx + C_2, \]

where \( C_2 \) is another constant of integration.

9. **Solve for \( y \)**:
Finally, we solve for \( y \):

\[ y = \frac{\int e^{\int f(x) dx} r(x) dx + C_2}{e^{\int f(x) dx}}. \]

This can be simplified to:

\[ y = \int e^{\int f(x) dx} r(x) dx \cdot \frac{1}{e^{\int f(x) dx}} + \frac{C_2}{e^{\int f(x) dx}}. \]

### Final Solution:
Thus, the solution of the differential equation

\[ \frac{dy}{dx} + f(x) y = r(x) \]

is given by:

\[ y = \frac{\int e^{\int f(x) dx} r(x) dx + C}{e^{\int f(x) dx}}. \]