Which one of the following is the differential equation that represents the family of curves `y=(1)/(2x^(2)-c)` where c is an arbitrary constant ?

Options

A. `(dy)/(dx)=4xy^(2)`
B. `(dy)/(dx)=1/y`
C. `(dy)/(dx)=x^(2)y`
D. `(dy)/(dx)=-4xy^(2)`

Answer

To find the differential equation that represents the family of curves given by the equation \( y = \frac{1}{2x^2 - c} \), where \( c \) is an arbitrary constant, we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the given equation**:
Start with the equation:
\[ y = \frac{1}{2x^2 - c} \]
We can rearrange this to express it in a different form:
\[ \frac{1}{y} = 2x^2 - c \]

2. **Differentiate both sides**:
Now, we will differentiate both sides with respect to \( x \). Using implicit differentiation, we have:
\[ \frac{d}{dx}\left(\frac{1}{y}\right) = \frac{d}{dx}(2x^2 - c) \]
The left-hand side can be differentiated using the chain rule:
\[ -\frac{1}{y^2} \frac{dy}{dx} = 4x \]
The right-hand side differentiates to \( 4x \) since the derivative of \( c \) (a constant) is \( 0 \).

3. **Solve for \(\frac{dy}{dx}\)**:
Rearranging the equation gives:
\[ -\frac{1}{y^2} \frac{dy}{dx} = 4x \]
Multiplying both sides by \( -y^2 \):
\[ \frac{dy}{dx} = -4xy^2 \]

4. **Final form of the differential equation**:
The resulting differential equation that represents the family of curves is:
\[ \frac{dy}{dx} = -4xy^2 \]

### Conclusion:
The differential equation that represents the family of curves \( y = \frac{1}{2x^2 - c} \) is:
\[ \frac{dy}{dx} = -4xy^2 \]