Question

If \[ y = a e^{bx} + c e^{dx} + x e^{bx} \] is the general solution of a differential equation, where \( a \) and \( c \) are arbitrary constants and \( b \) is a fixed constant, then the order of the differential equation is:

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

The order of a differential equation corresponds to the number of arbitrary constants in its general solution.

Answer 1

The Correct Option is A

Step 1: Understanding the given general solution 
We are given: \[ y = a e^{bx} + c e^{dx} + x e^{bx}. \] where \( a \) and \( c \) are arbitrary constants, and \( b \) is a fixed constant. 
Step 2: Determining the order of the differential equation 
The order of a differential equation is equal to the number of arbitrary constants in the general solution. In this case, the given solution contains two arbitrary constants: \( a \) and \( c \). Since we need to form a differential equation by eliminating these arbitrary constants, we differentiate successively. 
Step 3: Differentiating the given function 
Differentiating both sides with respect to \( x \): \[ y' = a b e^{bx} + c d e^{dx} + e^{bx} + b x e^{bx}. \] Differentiating again: \[ y'' = a b^2 e^{bx} + c d^2 e^{dx} + b e^{bx} + b e^{bx} + b^2 x e^{bx}. \] Since there are two arbitrary constants, differentiating twice is sufficient to eliminate them and obtain the required differential equation. Thus, the order of the differential equation is: \[ 2. \] 
Step 4: Conclusion 
Thus, the correct answer is: \[ 1. \]

Answer 2

To determine the order of the differential equation represented by the given general solution \(y = a e^{bx} + c e^{dx} + x e^{bx}\), we follow these steps:
1. A differential equation's order is defined as the highest derivative present in the equation.
2. The given solution involves three distinct terms: \(a e^{bx}\), \(c e^{dx}\), and \(x e^{bx}\). To construct the differential equation, we find derivatives of \(y\) until all arbitrary constants (in this case, \(a\) and \(c\)) can be eliminated.
3. First, compute the first derivative: \[ y' = \frac{d}{dx}(a e^{bx} + c e^{dx} + x e^{bx}) = abe^{bx} + cde^{dx} + e^{bx} + bxe^{bx} \]
4. Notice the first derivative contains arbitrary constants \(a\) and \(c\).
5. The appearance of the variable and its derivatives suggests that satisfying any distinct arbitrary constant can be achieved with the next differentiation degree.
6. Calculate the second derivative: \[ y'' = \frac{d}{dx}(abe^{bx} + bxe^{bx} + e^{bx} + cde^{dx}) = ab^2e^{bx} + be^{bx} + abxe^{bx} + b^2xe^{bx} + c^2e^{dx} \]
7. If necessary, continue differentiating until all arbitrary constants are eliminated. Generally, they are eliminated at the first derivative here; thus, indicating the minimal required differentiation is one (hence, one degree).

Therefore, the order of the differential equation is 1. This is because the highest required derivative to involve all arbitrary constants and fully form the differential equation after the necessary reduction and elimination is the first derivative.