The order & the degree of the differential equation whose general solution is, `y =c(x-c)^2`, are respectively

Options

A. 1,1
B. 1,2
C. 1,3
D. 2,1

Answer

To determine the order and degree of the differential equation whose general solution is given by \( y = c(x - c)^2 \), we can follow these steps:

### Step 1: Identify the given general solution
The general solution provided is:
\[ y = c(x - c)^2 \]

### Step 2: Differentiate the equation
To find the order of the differential equation, we need to differentiate the equation with respect to \( x \). Let's differentiate the equation once:
\[ \frac{dy}{dx} = y' = c \cdot 2(x - c) \cdot \frac{d}{dx}(x - c) = 2c(x - c) \]

### Step 3: Express \( c \) in terms of \( y \) and \( y' \)
From the first derivative, we can express \( c \):
\[ y' = 2c(x - c) \implies c = \frac{y'}{2(x - c)} \]

### Step 4: Substitute \( c \) back into the original equation
Now, we can substitute \( c \) back into the original equation:
\[ y = \frac{y'}{2(x - c)}(x - c)^2 \]

### Step 5: Rearranging the equation
Rearranging the equation gives us a relationship involving \( y \), \( y' \), and \( x \):
\[ y' = \frac{2y(x - c)}{(x - c)^2} \]

### Step 6: Differentiate again to find the second derivative
Next, we differentiate again to find the second derivative:
\[ \frac{d^2y}{dx^2} = y'' \]
This process can be repeated until we determine the highest order of differentiation needed to express \( y \) in terms of \( x \) and its derivatives.

### Step 7: Determine the order and degree
After performing the necessary differentiations, we find that the highest derivative we need to express the relationship is the third derivative. Thus, the order of the differential equation is 3.

The degree of the differential equation is determined by the highest power of the highest order derivative when the equation is expressed as a polynomial in derivatives. In this case, since we are differentiating three times and the equation can be expressed as a polynomial, the degree is also 3.

### Final Answer
Thus, the order and degree of the differential equation are:
- **Order:** 3
- **Degree:** 3

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