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

Options

A. `x^2+(y/x)=c`
B. `y^2+(y/x)=c`
C. `f(y/x)=cx`
D. `f(y/x)=cy`

Answer

To solve the differential equation

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

we can follow these steps:

### Step 1: Substitute \( v = \frac{y}{x} \)

Let \( v = \frac{y}{x} \). Then, we can express \( y \) in terms of \( v \) and \( x \):

\[ y = vx. \]

### Step 2: Differentiate \( y \)

Now, differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = v + x\frac{dv}{dx}. \]

### Step 3: Substitute into the differential equation

Now, substitute \( \frac{dy}{dx} \) and \( \frac{y}{x} \) into the original differential equation:

\[ v + x\frac{dv}{dx} = v + \frac{f(v)}{f'(v)}. \]

### Step 4: Simplify the equation

Subtract \( v \) from both sides:

\[ x\frac{dv}{dx} = \frac{f(v)}{f'(v)}. \]

### Step 5: Separate the variables

Now, we can separate the variables:

\[ \frac{f'(v)}{f(v)} dv = \frac{dx}{x}. \]

### Step 6: Integrate both sides

Integrate both sides:

\[ \int \frac{f'(v)}{f(v)} dv = \int \frac{dx}{x}. \]

The left side integrates to \( \log |f(v)| \) and the right side integrates to \( \log |x| + C \):

\[ \log |f(v)| = \log |x| + C. \]

### Step 7: Exponentiate both sides

Exponentiating both sides gives:

\[ f(v) = kx, \]

where \( k = e^C \).

### Step 8: Substitute back for \( v \)

Recall that \( v = \frac{y}{x} \), so we substitute back:

\[ f\left(\frac{y}{x}\right) = kx. \]

### Final Solution

Thus, the solution of the differential equation is:

\[ f\left(\frac{y}{x}\right) = cx, \]

where \( c \) is a constant.

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