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

The Correct Option is B

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