Write order of the differential equation of the family of following curves
`y= a cos (ax+b)`

Answer

To determine the order of the differential equation for the family of curves given by \( y = a \cos(ax + b) \), we will follow these steps:

### Step 1: Identify the given equation
The given equation is:
\[ y = a \cos(ax + b) \]
Here, \( a \) and \( b \) are constants.

### Step 2: Differentiate the equation with respect to \( x \)
We will find the first derivative of \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(a \cos(ax + b)) = -a^2 \sin(ax + b) \]
This result comes from applying the chain rule, where the derivative of \( \cos \) is \( -\sin \) and we multiply by the derivative of the inner function \( ax + b \), which is \( a \).

### Step 3: Differentiate again to find the second derivative
Next, we differentiate \( \frac{dy}{dx} \) to find the second derivative:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(-a^2 \sin(ax + b)) = -a^3 \cos(ax + b) \]
Again, we apply the chain rule, where the derivative of \( \sin \) is \( \cos \) and we multiply by \( a \).

### Step 4: Express the second derivative in terms of \( y \)
We know from the original equation that \( y = a \cos(ax + b) \). Thus, we can substitute this into our expression for the second derivative:
\[ \frac{d^2y}{dx^2} = -a^3 \frac{y}{a} = -a^2 y \]
This gives us a relationship between the second derivative and \( y \).

### Step 5: Formulate the differential equation
From the relationship we derived, we can write the differential equation as:
\[ \frac{d^2y}{dx^2} + a^2 y = 0 \]

### Step 6: Determine the order of the differential equation
The order of a differential equation is defined as the highest order of derivative present in the equation. In our case, the highest derivative is \( \frac{d^2y}{dx^2} \), which is of order 2.

### Conclusion
Thus, the order of the differential equation for the family of curves \( y = a \cos(ax + b) \) is:
\[ \text{Order} = 2 \]

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