A differential equation of the family of curves `y= a sin px + b cos px` where a, b are arbitrary constants, is given by

Options

A. `(d^(2)y)/(dx^(2))+py=0`
B. `(d^(2)y)/(dx^(2)) +p^(2)y=0`
C. `(d^(2)y)/(dx^(2)) -p^(2)y=0`
D. `(d^(2)y)/(dx^(2))-py=0`

Answer

To find the differential equation of the family of curves given by \( y = a \sin(px) + b \cos(px) \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the function:
\[ y = a \sin(px) + b \cos(px) \]
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = a \cdot p \cos(px) - b \cdot p \sin(px) \]
This can be simplified to:
\[ y' = p(a \cos(px) - b \sin(px)) \]

### Step 2: Differentiate again to find the second derivative
Next, we differentiate \( y' \) to find the second derivative:
\[ \frac{d^2y}{dx^2} = p \left( -a p \sin(px) - b p \cos(px) \right) \]
This simplifies to:
\[ \frac{d^2y}{dx^2} = -p^2(a \sin(px) + b \cos(px)) \]
Notice that \( a \sin(px) + b \cos(px) = y \), so we can rewrite the equation as:
\[ \frac{d^2y}{dx^2} = -p^2 y \]

### Step 3: Rearranging the equation
Rearranging the equation gives us:
\[ \frac{d^2y}{dx^2} + p^2 y = 0 \]

### Final Result
Thus, the differential equation of the family of curves \( y = a \sin(px) + b \cos(px) \) is:
\[ \frac{d^2y}{dx^2} + p^2 y = 0 \]