Form differential equation for `x^(2)=4by`

Options

A. `y=2xy_(1)`
B. `xy_(1)=2y`
C. `xy=2y_(1)`
D. `yy_(1)=2x`

Answer

To form the differential equation from the equation \( x^2 = 4By \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation:
\[ x^2 = 4By \]
We differentiate both sides with respect to \( x \).

### Step 2: Apply differentiation
Differentiating the left side:
\[ \frac{d}{dx}(x^2) = 2x \]
For the right side, since \( B \) is a constant, we apply the product rule:
\[ \frac{d}{dx}(4By) = 4B \frac{dy}{dx} \]
Thus, we have:
\[ 2x = 4B \frac{dy}{dx} \]

### Step 3: Express \(\frac{dy}{dx}\)
Rearranging the equation gives:
\[ \frac{dy}{dx} = \frac{2x}{4B} = \frac{x}{2B} \]

### Step 4: Substitute \( B \)
From the original equation \( x^2 = 4By \), we can express \( B \) in terms of \( x \) and \( y \):
\[ B = \frac{x^2}{4y} \]
Substituting this into the expression for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{x}{2 \left(\frac{x^2}{4y}\right)} = \frac{x \cdot 4y}{2x^2} = \frac{2y}{x} \]

### Step 5: Rearranging to form the differential equation
Now, we can rearrange this to form the required differential equation:
\[ x \frac{dy}{dx} = 2y \]

### Final Result
The differential equation formed from the given equation \( x^2 = 4By \) is:
\[ x \frac{dy}{dx} - 2y = 0 \]