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 \( y = a e^{(bx + c)} \), we will follow these steps:

### Step 1: Differentiate the given equation
We start with the general solution:
\[ y = a e^{(bx + c)} \]
Differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = a \cdot \frac{d}{dx}(e^{(bx + c)}) \]
Using the chain rule, we get:
\[ \frac{dy}{dx} = a e^{(bx + c)} \cdot b \]
This simplifies to:
\[ \frac{dy}{dx} = b y \]

### Step 2: Differentiate again
Now, we differentiate \( \frac{dy}{dx} = b y \) with respect to \( x \):
\[ \frac{d^2y}{dx^2} = b \cdot \frac{dy}{dx} \]
Substituting \( \frac{dy}{dx} = b y \) into this equation gives:
\[ \frac{d^2y}{dx^2} = b \cdot (b y) = b^2 y \]

### Step 3: Rearranging the equation
We can rearrange this to form a standard differential equation:
\[ \frac{d^2y}{dx^2} - b^2 y = 0 \]

### Step 4: 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, the highest derivative is \( \frac{d^2y}{dx^2} \), which is a second-order derivative.

### Conclusion
Thus, the order of the differential equation is:
\[ \text{Order} = 2 \]

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