Write order of the differential equation of the family of following curves
`y= a+be^(x+c)`

Answer

To find the order of the differential equation of the family of curves given by \( y = a + b e^{(x+c)} \), we will follow these steps:

### Step 1: Identify the constants in the equation
The equation has three constants: \( a \), \( b \), and \( c \).

### Step 2: Differentiate the equation with respect to \( x \)
We differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(a + b e^{(x+c)}) = b e^{(x+c)} \]
Here, \( a \) is a constant and its derivative is 0.

### Step 3: Express \( b \) in terms of \( y \)
From the first derivative, we can express \( b \):
\[ b = \frac{dy}{dx} e^{-(x+c)} \]
This still has the constant \( c \) in it.

### Step 4: Differentiate again to eliminate another constant
Now, we differentiate \( \frac{dy}{dx} \) again with respect to \( x \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(b e^{(x+c)}) = b e^{(x+c)} \]
Using the expression for \( b \) from the previous step, we can substitute it back.

### Step 5: Analyze the derivatives
Now we have:
- First derivative: \( \frac{dy}{dx} = b e^{(x+c)} \)
- Second derivative: \( \frac{d^2y}{dx^2} = b e^{(x+c)} \)

### Step 6: Determine the order of the differential equation
The highest derivative we have is the second derivative \( \frac{d^2y}{dx^2} \). Therefore, the order of the differential equation is 2.

### Conclusion
Thus, the order of the differential equation of the family of curves \( y = a + b e^{(x+c)} \) is **2**.

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