The substituting `y=vx` reduces the homogeneous differential equation `(dy)/(dx)=(y)/(x)+tan.(y)/(x)` to the form

Options

A. `(tan v)dv=xdx`
B. `(tanv)dv=(dx)/(x)`
C. `(cotv)dv=xdx`
D. `cotvdv =(dx)/(x)`

Explanation

To solve the given problem, we need to reduce the homogeneous differential equation

\[ \frac{dy}{dx} = \frac{y}{x} + \tan\left(\frac{y}{x}\right) \]

by substituting \( y = vx \), where \( v \) is a function of \( x \).

### Step-by-Step Solution:

1. **Substitution**:
We start by substituting \( y = vx \).

2. **Differentiate**:
Differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]

3. **Express \( \frac{y}{x} \)**:
From the substitution, we can express \( \frac{y}{x} \) as:
\[ \frac{y}{x} = v \]

4. **Substitute into the original equation**:
Now substitute \( \frac{dy}{dx} \) and \( \frac{y}{x} \) into the original differential equation:
\[ v + x \frac{dv}{dx} = v + \tan(v) \]

5. **Simplify the equation**:
We can cancel \( v \) from both sides:
\[ x \frac{dv}{dx} = \tan(v) \]

6. **Rearranging the equation**:
Rearranging gives:
\[ \frac{dv}{\tan(v)} = \frac{dx}{x} \]

7. **Using trigonometric identity**:
We know that \( \frac{1}{\tan(v)} = \cot(v) \), so we can write:
\[ \cot(v) dv = \frac{dx}{x} \]

### Final Result:
Thus, the homogeneous differential equation reduces to the form:
\[ \cot(v) dv = \frac{dx}{x} \]