The differential equation of ` y = c^(2) + c/x` is

Options

A. `x^(4) ((dy)/(dx))^(2) - x(dy)/(dx) =y`
B. `(d^(2)y)/(dx^(2)) +x (dy)/(dx) + y=0`
C. ` x^(3)((dy)/(dx))^(2) + x(dy)/(dx) = y`
D. `(d^(2)y)/(dx^(2)) + (dy)/(dx) -y=0`

Answer

To find the differential equation of the given function \( y = c^2 + \frac{c}{x} \), we will follow these steps:

### Step 1: Differentiate the equation with respect to \( x \)
Given the equation:
\[ y = c^2 + \frac{c}{x} \]
We differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = 0 + \frac{d}{dx}\left(\frac{c}{x}\right) \]
Using the quotient rule, we get:
\[ \frac{dy}{dx} = -\frac{c}{x^2} \]

### Step 2: Express \( c \) in terms of \( y \) and \( x \)
From the original equation, we can isolate \( c \):
\[ c = x(y - c^2) \]

### Step 3: Substitute \( c \) back into the derivative
Now we substitute \( c \) in the derivative equation:
\[ \frac{dy}{dx} = -\frac{x(y - c^2)}{x^2} = -\frac{y - c^2}{x} \]
This gives us:
\[ \frac{dy}{dx} = -\frac{y - c^2}{x} \]

### Step 4: Rearranging the equation
Rearranging the equation gives:
\[ x \frac{dy}{dx} + y = c^2 \]

### Step 5: Form the differential equation
Now we need to eliminate the constant \( c \). From the previous steps, we have:
\[ c^2 = x \frac{dy}{dx} + y \]
Thus, we can express the differential equation as:
\[ x \frac{dy}{dx} + y - c^2 = 0 \]

### Final Differential Equation
The final form of the differential equation is:
\[ x \frac{dy}{dx} + y = c^2 \]