The solution of the differential equation `x(dy)/(dx)=y ln ((y^(2))/(x^(2)))` is (where, c is an arbitrary constant)

Options

A. `y=x.e^(cx+1)`
B. `y=x.e^(cx-1)`
C. `y=x^(2).e^(cx+1)`
D. `y=x.e^(cx^(2)+(1)/(2))`

Answer

To solve the differential equation \( x \frac{dy}{dx} = y \ln\left(\frac{y^2}{x^2}\right) \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ x \frac{dy}{dx} = y \ln\left(\frac{y^2}{x^2}\right) \]
This can be rewritten as:
\[ x \frac{dy}{dx} = y \ln\left(y^2\right) - y \ln\left(x^2\right) \]
Using properties of logarithms, we simplify to:
\[ x \frac{dy}{dx} = 2y \ln(y) - 2y \ln(x) \]

### Step 2: Separate variables
We can separate the variables by dividing both sides by \(y\) and rearranging:
\[ \frac{dy}{2 \ln(y) - 2 \ln(x)} = \frac{dx}{x} \]
This simplifies to:
\[ \frac{dy}{2(\ln(y) - \ln(x))} = \frac{dx}{x} \]

### Step 3: Integrate both sides
Next, we integrate both sides:
\[ \int \frac{dy}{2(\ln(y) - \ln(x))} = \int \frac{dx}{x} \]
Let \(v = \frac{y}{x}\), then \(y = vx\) and \(\frac{dy}{dx} = v + x\frac{dv}{dx}\). Substitute \(y\) into the left side:
\[ \int \frac{1}{2(\ln(v) + \ln(x) - \ln(x))} dy = \int \frac{dx}{x} \]
This leads to:
\[ \int \frac{dy}{2 \ln(v)} = \ln|x| + C \]

### Step 4: Solve for \(v\)
After integrating, we can express the left-hand side in terms of \(v\):
\[ \frac{1}{2} \ln(\ln(v)) = \ln|x| + C \]
Exponentiating both sides gives:
\[ \ln(v) = e^{2(\ln|x| + C)} \]
This simplifies to:
\[ v = e^{2C} |x|^2 \]
Let \(k = e^{2C}\), then:
\[ \frac{y}{x} = kx^2 \]
Thus, we have:
\[ y = kx^3 \]

### Step 5: General solution
The general solution of the differential equation is:
\[ y = Cx^3 \]
where \(C\) is an arbitrary constant.

### Final Answer
The solution of the differential equation is:
\[ y = Cx^3 \]