If `(y^(3)-2x^(2)y)dx+(2xy^(2)-x^(3))dy=0,` then the value of `xysqrt(y^(2)-x^(2)),` is

Options

A. `y^(2)+x`
B. `xy^(2)`
C. any contant
D. None of these

Answer

To solve the differential equation given by:

\[ (y^3 - 2x^2y)dx + (2xy^2 - x^3)dy = 0, \]

we will follow these steps:

### Step 1: Rearranging the Equation
We can rearrange the equation into the form of \(\frac{dy}{dx}\):

\[ (2xy^2 - x^3)dy = -(y^3 - 2x^2y)dx. \]

Dividing both sides by \(dx\) and \(2xy^2 - x^3\):

\[ \frac{dy}{dx} = -\frac{y^3 - 2x^2y}{2xy^2 - x^3}. \]

### Step 2: Substituting \(y = vx\)
Let \(y = vx\), where \(v\) is a function of \(x\). Then, we have:

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

Substituting \(y\) into the equation gives:

\[ \frac{dy}{dx} = v + x\frac{dv}{dx} = -\frac{(vx)^3 - 2x^2(vx)}{2x(vx)^2 - x^3}. \]

### Step 3: Simplifying the Equation
Substituting \(y = vx\) into the right side:

\[ = -\frac{v^3x^3 - 2vx^3}{2vx^3 - x^3} = -\frac{x^3(v^3 - 2v)}{x^3(2v - 1)} = -\frac{v^3 - 2v}{2v - 1}. \]

Thus, we have:

\[ v + x\frac{dv}{dx} = -\frac{v^3 - 2v}{2v - 1}. \]

### Step 4: Rearranging the Equation
Rearranging gives:

\[ x\frac{dv}{dx} = -\frac{v^3 - 2v}{2v - 1} - v. \]

### Step 5: Finding a Common Denominator
Combining the fractions:

\[ x\frac{dv}{dx} = -\frac{v^3 - 2v + v(2v - 1)}{2v - 1} = -\frac{v^3 + v^2 - 2v}{2v - 1}. \]

### Step 6: Separating Variables
Now, we can separate variables:

\[ \frac{2v - 1}{v^3 + v^2 - 2v} dv = -\frac{1}{x} dx. \]

### Step 7: Integrating Both Sides
Integrate both sides:

\[ \int \frac{2v - 1}{v^3 + v^2 - 2v} dv = -\int \frac{1}{x} dx. \]

### Step 8: Solving the Integrals
The left side can be solved using partial fractions, and the right side gives:

\[ -\ln|x| + C. \]

### Step 9: Back Substituting for \(y\)
After solving the integral, we substitute back \(v = \frac{y}{x}\) to find the relation between \(y\) and \(x\).

### Step 10: Finding \(x y \sqrt{y^2 - x^2}\)
From the derived relation, we can express \(xy\sqrt{y^2 - x^2}\) in terms of a constant \(C\):

\[ xy\sqrt{y^2 - x^2} = C. \]

### Conclusion
Thus, the value of \(xy\sqrt{y^2 - x^2}\) is a constant \(C\).

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