Question

The solution of the differential equation \(\frac{d^3y}{dx^3} + 3\frac{d^2y}{dx^2} + 2\frac{dy}{dx} = 0\) is

• A. \(y = a + be^{-x} + ce^{-2x}\)
• B. \(y = a + be^x + ce^{2x}\)
• C. \(y = ae^{-x} + be^{-2x} + ce^x\)
• D. \(y = a + be^{-2x} + ce^{-3x}\)

Hint:

For homogeneous linear DEs with constant coefficients, the process is always the same: form the auxiliary equation, find its roots, and write the solution based on the type of roots. A root of \(m=0\) always contributes a simple constant term to the solution.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
Find PI of (D²+3D+2)y=e⁻²ˣ.

Step 2: Key Formula (Alternate):
PI = (1/f(D))eᵃˣ. If f(a)=0, use x(1/f'(a))eᵃˣ.

Step 3: Detailed Explanation:
a=-2. f(-2)=4-6+2=0 (failure). f'(D)=2D+3, f'(-2)=-1. PI = x·(1/-1)·e⁻²ˣ = -xe⁻²ˣ.

Step 4: Final Answer:
PI is -xe⁻²ˣ.