Question

Let $ a $ and $ b $ be arbitrary constants and $ C $ be a fixed constant. If $ y = ae^{2x} + bxe^{2x} + C $ is the general solution of a differential equation, then the order of that differential equation is:

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

Hint:

When working with differential equations, the order is determined by the highest derivative required to express the solution without arbitrary constants.

Answer 1

The Correct Option is B

We are given the general solution: \[ y = ae^{2x} + bxe^{2x} + C \] Step 1: Differentiate the given solution.
First, find the first derivative of \( y \) with respect to \( x \): \[ \frac{dy}{dx} = 2ae^{2x} + be^{2x} + 2bxe^{2x} \] Step 2: Differentiate again.
Next, find the second derivative of \( y \): \[ \frac{d^2y}{dx^2} = 4ae^{2x} + 4be^{2x} + 4bxe^{2x} \] Step 3: Analyze the order of the differential equation.
Since we need to eliminate the arbitrary constants \( a \), \( b \), and \( C \), the highest derivative we need to take is the second derivative, which is the second order differential equation.
Thus, the order of the differential equation is 2.
Thus, the correct answer is: \[ \boxed{2} \]