Question

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

• A. \( y(x) = \sin x - \int_0^x \sin(x - t) g(t) \, dt \)
• B. \( y(x) = \sin x + \int_0^x \sin(x - t) g(t) \, dt \)
• C. \( y(x) = \sin x + \int_0^x \cos(x - t) g(t) \, dt \)
• D. \( y(x) = \sin x + \int_0^x \cos(x - t) g(t) \, dt \)

Hint:

For second-order linear differential equations, the general solution is the sum of the homogeneous solution and a particular solution. The particular solution is often found using methods like undetermined coefficients or variation of parameters.

Answer 1

The Correct Option is B

We are given the second-order differential equation: \[ \frac{d^2 y}{dx^2} + y = g(x), \] and the initial conditions \( y(0) = 0 \), \( y'(0) = 1 \). To solve this, we first solve the homogeneous equation \( \frac{d^2 y}{dx^2} + y = 0 \). The general solution to this homogeneous equation is: \[ y_h(x) = A \sin x + B \cos x, \] where \( A \) and \( B \) are constants to be determined by initial conditions. Next, we find a particular solution to the non-homogeneous equation. Using the method of undetermined coefficients or variation of parameters, we can propose that the particular solution has the form: \[ y_p(x) = \int_0^x \sin(x - t) g(t) \, dt. \] Thus, the complete solution is: \[ y(x) = \sin x + \int_0^x \sin(x - t) g(t) \, dt. \] Now, applying the initial conditions: - \( y(0) = 0 \) gives the value of the constant \( B \). - \( y'(0) = 1 \) gives the value of the constant \( A \). Therefore, the correct solution is option (B).