The differential equation of the family of curves `y=k_(1)x^(2)+k_(2)` is given by (where, `k_(1) and k_(2)` are arbitrary constants and `y_(1)=(dy)/(dx), y_(2)=(d^(2)y)/(dx^(2))`)

Options

A. `y_(1)=x^(2)y_(2)`
B. `(y_(1))^(2)=xy_(2)`
C. `xy_(2)=y_(1)`
D. `y_(1)y_(2)=x`

Answer

To find the differential equation of the family of curves given by \( y = k_1 x^2 + k_2 \), where \( k_1 \) and \( k_2 \) are arbitrary constants, we will follow these steps:

### Step 1: Differentiate the equation with respect to \( x \)
We start with the equation:
\[ y = k_1 x^2 + k_2 \]
Now, we differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(k_1 x^2 + k_2) \]
Using the power rule for differentiation, we get:
\[ \frac{dy}{dx} = 2k_1 x + 0 = 2k_1 x \]
Let’s denote \( \frac{dy}{dx} \) as \( y_1 \):
\[ y_1 = 2k_1 x \]

### Step 2: Differentiate again to find the second derivative
Next, we differentiate \( y_1 \) with respect to \( x \) to find \( y_2 \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(2k_1 x) \]
This gives us:
\[ \frac{d^2y}{dx^2} = 2k_1 \]
Let’s denote \( \frac{d^2y}{dx^2} \) as \( y_2 \):
\[ y_2 = 2k_1 \]

### Step 3: Express \( k_1 \) in terms of \( y_2 \)
From the equation \( y_2 = 2k_1 \), we can express \( k_1 \) as:
\[ k_1 = \frac{y_2}{2} \]

### Step 4: Substitute \( k_1 \) back into the equation for \( y_1 \)
Now, we substitute \( k_1 \) back into the equation for \( y_1 \):
\[ y_1 = 2k_1 x = 2\left(\frac{y_2}{2}\right)x = y_2 x \]

### Step 5: Formulate the differential equation
Now we have:
\[ y_1 = y_2 x \]
This is the required differential equation for the family of curves.

### Final Answer
The differential equation of the family of curves \( y = k_1 x^2 + k_2 \) is:
\[ \frac{dy}{dx} = \frac{d^2y}{dx^2} \cdot x \]

---