Differential equation of the function `c + 4yx = 0` is

Options

A. `xy + (dy)/(dx) = 0`
B. `x(dy)/(dx) + y = 0`
C. `(dy)/(dx) - 4xy = 0`
D. `x(dy)/(dx) + 1 = 0`

Answer

To find the differential equation of the function \( c + 4yx = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the given equation to isolate \( y \):
\[ 4yx = -c \]
This can be rewritten as:
\[ yx = -\frac{c}{4} \]

### Step 2: Differentiate with Respect to \( x \)
Now, we differentiate both sides of the equation with respect to \( x \). Since \( c \) is a constant, the derivative of \( -\frac{c}{4} \) will be 0. We apply the product rule on the left side:
\[ \frac{d}{dx}(yx) = \frac{d}{dx}(-\frac{c}{4}) = 0 \]
Using the product rule:
\[ x \frac{dy}{dx} + y \frac{d}{dx}(x) = 0 \]
This simplifies to:
\[ x \frac{dy}{dx} + y = 0 \]

### Step 3: Write the Differential Equation
The resulting equation from the differentiation gives us the required differential equation:
\[ x \frac{dy}{dx} + y = 0 \]

### Final Result
Thus, the differential equation of the function \( c + 4yx = 0 \) is:
\[ x \frac{dy}{dx} + y = 0 \]

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