The differential equation whose solution is `(x - h)^2 + (y-k)^2= a^2` ( a is given constatnt) is :

Options

A. ` { 1+ ((dy)/(dx))^2}^a= a^2 ( d^2 y)/(dx ^2)`
B. ` { 1+((dy)/(dx))^2}^2=a^2((d^2 y)/(dx^2))^2`
C. ` {1 +((dy)/(dx ) ) }^2 = a^2 ( (d^2 y) /( dx^2))^2`
D. none of these

Answer

To find the differential equation whose solution is given by the equation \((x - h)^2 + (y - k)^2 = a^2\), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the equation:
\[ (x - h)^2 + (y - k)^2 = a^2 \]
Differentiating both sides with respect to \(x\):
\[ \frac{d}{dx}[(x - h)^2] + \frac{d}{dx}[(y - k)^2] = \frac{d}{dx}[a^2] \]
Since \(a\) is a constant, the derivative of \(a^2\) is 0. Therefore, we have:
\[ 2(x - h) + 2(y - k) \frac{dy}{dx} = 0 \]

### Step 2: Rearrange the equation
We can simplify the equation:
\[ 2(x - h) + 2(y - k) \frac{dy}{dx} = 0 \]
Dividing through by 2:
\[ (x - h) + (y - k) \frac{dy}{dx} = 0 \]
Rearranging gives us:
\[ (y - k) \frac{dy}{dx} = - (x - h) \]

### Step 3: Differentiate again
Now, we differentiate the equation we obtained:
\[ \frac{d}{dx}[(y - k) \frac{dy}{dx}] = \frac{d}{dx}[-(x - h)] \]
Using the product rule on the left side:
\[ \frac{dy}{dx} \cdot \frac{dy}{dx} + (y - k) \frac{d^2y}{dx^2} = -1 \]
This simplifies to:
\[ \left(\frac{dy}{dx}\right)^2 + (y - k) \frac{d^2y}{dx^2} = -1 \]

### Step 4: Substitute \(y - k\) and \(\frac{dy}{dx}\)
Let \(y' = \frac{dy}{dx}\) and \(y'' = \frac{d^2y}{dx^2}\):
\[ y'^{2} + (y - k) y'' = -1 \]

### Final Differential Equation
Thus, the final form of the differential equation is:
\[ y''(y - k) + (y')^2 + 1 = 0 \]