91Ó°ÊÓ

A homogeneous equation is a type of differential equation where the right-hand side equals zero. In this exercise, the homogeneous equation is given by \[ y'' + 9y = 0. \]This form indicates that all changes in our unknown function, \(y\), are determined solely by the function and its derivatives. Solving this type of equation involves finding the roots of the associated characteristic equation.
The general solution for the homogeneous equation involves functions like sine and cosine, especially when the roots are complex numbers. This is due to the exponential properties of the complex numbers' components. Here, with roots \(\pm 3i\), the solution is \[ y_c(t) = C_1 \cos(3t) + C_2 \sin(3t), \]where \(C_1\) and \(C_2\) are constants determined by initial conditions.

The characteristic equation is derived from the homogeneous differential equation by replacing the derivative operators with powers of a variable, usually \(r\). For our equation, \[ y'' + 9y = 0, \]we substitute the form \(y = e^{rt}\), resulting in the characteristic equation \[ r^2 + 9 = 0. \]Solving this equation for \(r\) gives the values that dictate the nature of the solution to the differential equation. In this case, solving \[ r^2 + 9 = 0 \]results in complex roots \(r = \pm 3i\). This means the solution will have oscillatory components, identifiable as sine and cosine terms with frequencies related to these imaginary parts.

Initial conditions are specific values that define the state of the system at a particular instance, usually at the start (\(t = 0\)). These conditions allow us to determine the constants present in a general solution. For the exercise, the initial conditions provided are

• \(y(0) = 1\)
• \(y'(0) = 0\)

Using the general solution \[ y_c(t) = C_1 \cos(3t) + C_2 \sin(3t), \]we apply the conditions to establish specific values for \(C_1\) and \(C_2\). However, to solve this completely, the particular solution is needed as well, which depends on \(g(t)\) that is not provided here.

A particular solution of a differential equation accounts for the non-homogeneous part, the function \(g(t)\). This solution complements the general solution from the homogeneous equation to account for all terms in the original equation. When \(g(t)\) is given, a suitable function form is postulated—for example, polynomials, exponential, or trigonometric functions—and coefficients are determined by substituting this form back into the original differential equation.
In this specific exercise, without knowing \(g(t)\), a particular solution cannot be explicitly determined. Once \(g(t)\) is provided, this particular solution becomes possible to find a complete solution combining it with the homogeneous part, resulting in the equation \[ y(t) = y_c(t) + y_p(t). \]