.Let `y=y(x)` be the solution of the differential equation `(3y^(2)-5x^(2))ydx+2x(x^(2)-y^(2))dy=0` such that `y(1)=1`.Then `abs((y(2))^(3)-12y(2))` is equal to :

Options

A. `64`
B. `32sqrt(2)`
C. `32`
D. `16sqrt(2)`

Answer

To solve the given differential equation and find the value of \( |(y(2))^3 - 12y(2)| \), we will follow these steps:

### Step 1: Rewrite the Differential Equation
The given differential equation is:
\[ (3y^2 - 5x^2) y \, dx + 2x (x^2 - y^2) \, dy = 0 \]
We can rearrange this to:
\[ \frac{dy}{dx} = -\frac{(3y^2 - 5x^2) y}{2x (x^2 - y^2)} \]

### Step 2: Substitute \( y = vx \)
Let \( y = vx \), where \( v \) is a function of \( x \). Then, we have:
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]
Substituting \( y = vx \) into the differential equation gives:
\[ \frac{dy}{dx} = -\frac{(3(vx)^2 - 5x^2)(vx)}{2x(x^2 - (vx)^2)} \]
This simplifies to:
\[ v + x \frac{dv}{dx} = -\frac{(3v^2 - 5)v}{2(1 - v^2)} \]

### Step 3: Rearranging the Equation
Rearranging gives:
\[ x \frac{dv}{dx} = -\frac{(3v^2 - 5)v}{2(1 - v^2)} - v \]
Combining terms leads to:
\[ x \frac{dv}{dx} = -\frac{(3v^2 - 5)v + 2v(1 - v^2)}{2(1 - v^2)} \]
This can be simplified further.

### Step 4: Separate Variables
We can separate the variables:
\[ \frac{2(1 - v^2)}{(3v^2 - 5)v + 2v(1 - v^2)} \, dv = -\frac{dx}{x} \]

### Step 5: Integrate Both Sides
Integrating both sides:
\[ \int \frac{2(1 - v^2)}{(3v^2 - 5)v + 2v(1 - v^2)} \, dv = -\int \frac{dx}{x} \]
This will yield a logarithmic function on the right side.

### Step 6: Solve for the Constant
Using the initial condition \( y(1) = 1 \) (thus \( v(1) = 1 \)), we can find the constant of integration.

### Step 7: Find \( y(2) \)
Substituting \( x = 2 \) into the equation we derived will give us \( y(2) \).

### Step 8: Calculate \( |(y(2))^3 - 12y(2)| \)
Finally, we compute \( |(y(2))^3 - 12y(2)| \).