Solving Nonhomogeneous Differential Equations with Trigonometric Functions

Given the differential equation $y'' + 4y = \, \sin x$, find the general solution.

To solve the differential equation $y'' + 4y = \sin x$, we need to find its general solution. This is a nonhomogeneous second-order linear differential equation with constant coefficients. A common strategy for solving such equations is to first find the complementary, or homogeneous, solution. This involves solving the associated homogeneous equation $y'' + 4y = 0$, which usually entails finding the roots of its characteristic equation. For this problem, the characteristic equation is typically a quadratic equation in terms of $y$'s coefficients, and the solution may involve a combination of exponential functions depending on whether the roots are real or complex.

After determining the complementary solution, we then tackle the particular solution that corresponds to the nonhomogeneous part, which is $\sin x$ in this instance. This often involves using methods such as undetermined coefficients or variation of parameters. The method of undetermined coefficients is particularly useful here as it allows us to guess a form for the particular solution based on the type of function on the right-hand side of the equation. For a sinusoidal function like $\sin x$, our guessed form typically involves sine and cosine terms. The key is to adjust the coefficients of these terms to satisfy the original differential equation. Once the particular solution is found, it is added to the complementary solution to form the general solution.

This approach not only helps address the given equation but also builds a foundational understanding of solving similar nonhomogeneous differential equations with trigonometric or exponential forcing functions.