Question

The solution of the second order differential equation \(\frac{d^2y}{dx^2}+2 \frac{dy}{dx} + y = 0\) with boundary conditions y(0) = 1 and y(1) = 3 is
1. \(e^{-x} +[3esin(\frac{\pi x}{2})-1]xe^{-x}\)
2. e^-x - (3e - 1) xe^-x
3. e^-x + (3e - 1) xe^-x
4. \(e^{-x} -[3esin(\frac{\pi x}{2})-1]xe^{-x}\)

Answer 1

Correct Answer - Option 3 : e^-x + (3e - 1) xe^-x

Explanation:

\(\frac{d^2y}{dx^2}+2 \frac{dy}{dx} + y = 0\)

(D² + 2D + 1)y = 0

y(0) = 1

y(1) = 3

Auxiliary equation is;

m² + 2m + 1 + 0

m = -1, -1

CF = (C₁+C₂x)e^-x

and PI = 0

Using boundary conditions;

y(0) = 1,  C₁ = 1

y(1) = 3,  C₂ = 3e - 1

So the solution of equation is;

y = CF + PI

y = (C1+C2x)e^-x + 0

y = {1 + (3e - 1 )x}e^-x

y = e^-x + (3e - 1) xe^-x