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

To solve the differential equation \(\frac{d^2y}{dx^2}+y=g(x)\) with the initial conditions \(\(y(0) = 0\)\) and \(\(y'(0) = 1\)\), we use the theory of linear differential equations and Green's functions.

The given differential equation is non-homogeneous. For such equations, we find the general solution as a sum of the complementary function (solution of the homogeneous equation) and a particular solution.

1. First, solve the homogeneous equation:

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

Its characteristic equation is \(r^2 + 1 = 0\), giving roots \(r = \pm i\).

The complementary function is:

\(y_c(x) = C_1 \cos x + C_2 \sin x\).

1. Particular Integral:

To find the particular solution using the method of undetermined coefficients or the variation of parameters, assume a particular solution as:

\(\int^x_0 \sin(x-t)g(t)dt\). 

1. Combine the complementary function and particular solution:

The general solution is:

\(y(x) = C_1 \cos x + C_2 \sin x + \int^x_0 \sin(x-t)g(t)dt\).

Apply initial conditions \(y(0) = 0\) and \(y'(0) = 1\):

1. Using \(y(0) = 0\):

\(C_1 \cdot 1 + C_2 \cdot 0 = 0\) → \(C_1 = 0\)

1. Using \(y'(0) = 1\):

First, find \(y'(x)\):

\(y'(x) = C_1 (-\sin x) + C_2 (\cos x) + \frac{d}{dx} \left( \int^x_0 \sin(x-t)g(t)dt \right)\)

\(= C_2 \cos x + \sin(x-t) g(t) \; |_{t=0}^{t=x}\)

Evaluate at \(x = 0\):

\(y'(0) = C_2 \cdot 1 + 0 = 1\) → \(C_2 = 1\)

1. Therefore, the solution becomes:

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

Thus, the correct option is:

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

This option aligns with the initial conditions and the form required to solve the given differential equation using standard methods for non-homogeneous linear differential equations with constant coefficients.