The general solution of a differential equation is `y= ae ^(bx+ c) ` where are arbitrary constants. The order the differential equation is :

Options

A. 3
B. 2
C. 1
D. none of these

Answer

To determine the order of the differential equation given the general solution \( y = ae^{bx+c} \), we will follow these steps:

### Step 1: Identify the general solution
The general solution of the differential equation is given as:
\[ y = ae^{bx+c} \]
where \( a \) and \( b \) are arbitrary constants.

### Step 2: Differentiate the equation
We will differentiate \( y \) with respect to \( x \) to find the first derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{d}{dx}(ae^{bx+c}) = a \cdot e^{bx+c} \cdot \frac{d}{dx}(bx+c) \]
Using the chain rule, we differentiate \( bx+c \):
\[ \frac{d}{dx}(bx+c) = b \]
Thus, the first derivative becomes:
\[ \frac{dy}{dx} = abe^{bx+c} \]

### Step 3: Differentiate again to find the second derivative
Next, we differentiate \( \frac{dy}{dx} \) to find the second derivative \( \frac{d^2y}{dx^2} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(abe^{bx+c}) = ab \cdot e^{bx+c} \cdot \frac{d}{dx}(bx+c) = ab \cdot e^{bx+c} \cdot b \]
Thus, the second derivative is:
\[ \frac{d^2y}{dx^2} = ab^2 e^{bx+c} \]

### Step 4: Analyze the derivatives
Now we have:
1. \( y = ae^{bx+c} \)
2. \( \frac{dy}{dx} = abe^{bx+c} \)
3. \( \frac{d^2y}{dx^2} = ab^2 e^{bx+c} \)

### Step 5: Determine the order of the differential equation
The order of a differential equation is defined as the highest derivative present in the equation. In our case, we have the first derivative \( \frac{dy}{dx} \) and the second derivative \( \frac{d^2y}{dx^2} \).

Since the highest derivative is the second derivative, the order of the differential equation is:
\[ \text{Order} = 2 \]

### Conclusion
Thus, the order of the differential equation is 2.

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