Find the equation of the curve in which the subnormal varies as the square of the ordinate.

Explanation

To find the equation of the curve in which the subnormal varies as the square of the ordinate, we will follow these steps:

### Step 1: Understand the terms
- The **ordinate** refers to the y-coordinate of a point on the curve, which is represented by \( y \).
- The **subnormal** is defined as \( y \frac{dy}{dx} \).

### Step 2: Set up the relationship
According to the problem, the subnormal varies as the square of the ordinate. This can be expressed mathematically as:
\[ y \frac{dy}{dx} \propto y^2 \]
This means:
\[ y \frac{dy}{dx} = k y^2 \]
where \( k \) is a constant of proportionality.

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

### Step 4: Separate the variables
To solve this differential equation, we can separate the variables:
\[ \frac{dy}{y} = k \, dx \]

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

### Step 6: Solve for \( y \)
To find \( y \), we exponentiate both sides:
\[ y = e^{kx + C} \]
This can be rewritten as:
\[ y = e^C e^{kx} \]
Let \( A = e^C \) (a new constant), so we have:
\[ y = A e^{kx} \]

### Final Equation
Thus, the equation of the curve is:
\[ y = A e^{kx} \]