The slope of the tangent at any arbitrary point of a curve is twice the product of the abscissa and square of the ordinate of the point. Then, the equation of the curve is (where c is an arbitrary constant)

Options

A. `x^(2)y+y+c=0`
B. `x^(2)y+cy+1=0`
C. `xy+y+c=0`
D. `xy^(2)+cy+y=0`

Answer

To solve the problem, we need to find the equation of the curve given that the slope of the tangent at any point is twice the product of the abscissa (x-coordinate) and the square of the ordinate (y-coordinate).

### Step-by-Step Solution:

1. **Understanding the Slope**:
The slope of the tangent to the curve at any point \((x, y)\) is given by \(\frac{dy}{dx}\). According to the problem, this slope is equal to \(2xy^2\).

\[ \frac{dy}{dx} = 2xy^2 \]

**Hint**: Remember that the slope of the tangent line at a point on a curve is represented by the derivative \(\frac{dy}{dx}\).

2. **Rearranging the Equation**:
We can rearrange the equation to separate the variables \(y\) and \(x\):

\[ \frac{dy}{y^2} = 2x \, dx \]

**Hint**: To separate variables, we want all terms involving \(y\) on one side and all terms involving \(x\) on the other side.

3. **Integrating Both Sides**:
Now, we integrate both sides. The left side involves integrating \(\frac{1}{y^2}\) and the right side involves integrating \(2x\):

\[ \int \frac{dy}{y^2} = \int 2x \, dx \]

The left side integrates to \(-\frac{1}{y}\) and the right side integrates to \(x^2 + C\):

\[ -\frac{1}{y} = x^2 + C \]

**Hint**: Remember to add the constant of integration \(C\) after integrating the right side.

4. **Rearranging the Equation**:
We can rearrange this equation to express \(y\) in terms of \(x\):

\[ -1 = x^2y + Cy \]

**Hint**: You can multiply both sides by \(-1\) to simplify the equation.

5. **Final Form of the Equation**:
Rearranging gives us the final equation of the curve:

\[ x^2y + Cy + 1 = 0 \]

**Hint**: This is the standard form of the equation of the curve, where \(C\) is an arbitrary constant.

### Conclusion:
Thus, the equation of the curve is:

\[ x^2y + Cy + 1 = 0 \]

This matches with the option provided in the problem statement.