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When solving a second-order linear homogeneous differential equation like \( y'' + 9y = 0 \), the process begins with finding the characteristic equation. This equation helps determine the behavior of possible solutions without directly solving the differential equation itself. The characteristic equation comes from assuming a solution of the form \( y = e^{rx} \), where \( r \) is a constant. By substituting \( y = e^{rx} \) into the differential equation \( y'' + 9y = 0 \), we get the characteristic equation: \[ r^2 + 9 = 0 \]Finding the roots of this characteristic equation is key to determining the form of the general solution of the differential equation. This step translates a complex differential problem into an algebraic one, making it significantly easier to solve.

When solving the characteristic equation \( r^2 + 9 = 0 \), you find that the roots are imaginary. Imaginary roots occur when the equation involves a negative value under the square root, resulting in the need for imaginary numbers. In our case, the roots are found by solving:\[ r^2 = -9 \]This gives:\[ r = \pm 3i \]where \( i \) is the imaginary unit, defined by \( i^2 = -1 \). Imaginary roots indicate that the solutions of the differential equation will involve trigonometric functions, specifically sine and cosine. This happens because the exponential function with an imaginary exponent can be expressed in terms of sine and cosine, according to Euler's formula.

With the roots of the characteristic equation \( r = \pm 3i \) determined as imaginary, we can now write the general solution of the differential equation. Imaginary roots lead to a solution in terms of trigonometric functions:\[ y(x) = C_1 \cos(3x) + C_2 \sin(3x) \]Here, \( C_1 \) and \( C_2 \) are arbitrary constants. These constants are crucial because they allow the solution to potentially match any initial conditions that may be given in particular problems.This form of the solution reflects the oscillatory nature of the functions sine and cosine, often appearing in systems with periodic behavior, like mechanical vibrations or alternating electric currents.