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: \[ 1. \]