Equation of the curve in which the subnormal is twice the square of the ordinate is given by

Options

A. `log y = 2x +log c`
B. `y=ce^(2x)`
C. `log y =2x^2 -log c`
D. None of these

Answer

To find the equation of the curve where the subnormal is twice the square of the ordinate, we can follow these steps:

### Step 1: Understand the concept of subnormal
The subnormal at any point on the curve is given by the formula:
\[ \text{Subnormal} = y \frac{dy}{dx} \]
where \( y \) is the ordinate (y-coordinate) and \( \frac{dy}{dx} \) is the slope of the tangent to the curve at that point.

### Step 2: Set up the equation
According to the problem, the subnormal is twice the square of the ordinate:
\[ y \frac{dy}{dx} = 2y^2 \]

### Step 3: Simplify the equation
We can simplify this equation by dividing both sides by \( y \) (assuming \( y \neq 0 \)):
\[ \frac{dy}{dx} = 2y \]

### Step 4: Separate the variables
Now, we can separate the variables:
\[ \frac{1}{y} dy = 2 dx \]

### Step 5: Integrate both sides
Next, we integrate both sides:
\[ \int \frac{1}{y} dy = \int 2 dx \]
This gives us:
\[ \ln |y| = 2x + C \]
where \( C \) is the constant of integration.

### Step 6: Solve for \( y \)
To solve for \( y \), we exponentiate both sides:
\[ |y| = e^{2x + C} = e^{2x} \cdot e^C \]
Let \( k = e^C \), then:
\[ y = k e^{2x} \]

### Step 7: General solution
Thus, the general solution for the equation of the curve is:
\[ y = C e^{2x} \]
where \( C \) is a constant.

### Step 8: Conclusion
The equation of the curve in which the subnormal is twice the square of the ordinate is:
\[ y = C e^{2x} \]