Question

Let 𝑔: ℝ → ℝ be a continuous function. Which one of the following is the solution of the differential equation
$\frac{d^2y}{dx^2}+y=g(x)\ \ \ \ \text{for}\ x \in \R$,
satisfying the conditions y(0) = 0, y'(0) = 1 ?

• A. $y(x)=\sin x-\int^x_0 \sin(x-t)g(t)dt$
• B. $y(x)=\sin x+\int^x_0 \sin(x-t)g(t)dt$
• C. $y(x)=\sin x-\int^x_0 \cos(x-t)g(t)dt$
• D. $y(x)=\sin x+\int^x_0 \cos(x-t)g(t)dt$

Answer 1

Answer: (B)

$y(x)=\sin x+\int^x_0 \sin(x-t)g(t)dt$

Answer 2

The correct option is (B) : $y(x)=\sin x+\int^x_0 \sin(x-t)g(t)dt$.