The solution of the differential equation
`(dy)/(dx) = (y)/(x)+ (Q((y)/(x)))/(Q'((y)/(x)))` is

Options

A. `Q((y)/(x)) = kx`
B. `x Q((y)/(x)) = k`
C. `Q((y)/(x)) = ky`
D. `y. Q((y)/(x)) = k`

Answer

To solve the differential equation

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

we will follow these steps:

### Step 1: Introduce a substitution
Let \( v = \frac{y}{x} \). Then, we can express \( y \) in terms of \( v \) and \( x \):
\[ y = vx. \]

### Step 2: Differentiate \( y \) with respect to \( x \)
Using the product rule, we differentiate \( y \):
\[ \frac{dy}{dx} = v + x \frac{dv}{dx}. \]

### Step 3: Substitute into the original equation
Now, substitute \( \frac{dy}{dx} \) and \( y \) into the original differential equation:
\[ v + x \frac{dv}{dx} = \frac{vx}{x} + \frac{Q(v)}{Q'(v)}. \]
This simplifies to:
\[ v + x \frac{dv}{dx} = v + \frac{Q(v)}{Q'(v)}. \]

### Step 4: Simplify the equation
Subtract \( v \) from both sides:
\[ x \frac{dv}{dx} = \frac{Q(v)}{Q'(v)}. \]

### Step 5: Separate variables
Rearranging gives us:
\[ \frac{Q'(v)}{Q(v)} dv = \frac{1}{x} dx. \]

### Step 6: Integrate both sides
Integrate both sides:
\[ \int \frac{Q'(v)}{Q(v)} dv = \int \frac{1}{x} dx. \]

### Step 7: Solve the integrals
The left side integrates to \( \ln |Q(v)| \) and the right side integrates to \( \ln |x| + C \):
\[ \ln |Q(v)| = \ln |x| + C. \]

### Step 8: Exponentiate to solve for \( Q(v) \)
Exponentiating both sides gives:
\[ |Q(v)| = k|x|, \quad \text{where } k = e^C. \]

### Step 9: Substitute back for \( y \)
Recall \( v = \frac{y}{x} \), so we can write:
\[ |Q\left(\frac{y}{x}\right)| = k|x|. \]

### Final Solution
Thus, the solution of the differential equation is given by:
\[ Q\left(\frac{y}{x}\right) = kx. \]