91Ó°ÊÓ

In the context of differential equations, the characteristic equation is a polynomial whose roots are directly related to the solutions of a differential equation. For a second-order linear differential equation like \( y'' + 9y = 0 \), we form the characteristic equation by assuming a solution of the form \( y = e^{mx} \), where \( m \) is a constant. Substituting this assumed solution back into the differential equation gives us a characteristic polynomial.

For the given differential equation, substituting in our assumed solution leads to the characteristic equation \( m^2 + 9 = 0 \). Solving this equation for \( m \) yields the roots \( m = \pm 3i \). These roots indicate the nature of solutions that we can expect.

• Real roots: Result in exponential solutions.
• Complex roots: Indicate oscillatory behavior, leading to sine and cosine terms.

With complex roots \( m = \pm 3i \), the general solution becomes a linear combination of sine and cosine functions: \( y(x) = c_1 \cos(3x) + c_2 \sin(3x) \). The constants \( c_1 \) and \( c_2 \) will be determined by initial conditions.

Initial conditions are constraints that help determine the specific solution to a differential equation problem. These conditions specify the value of the function or its derivatives at a particular point. Using initial conditions helps narrow down the infinite family of solutions of a differential equation to a single specific solution.

For the given problem, the initial conditions are \( y(0) = 1 \) and \( y'(0) = 1 \). These tell us the exact values of the function and its derivative at \( x = 0 \). Applying these conditions:

• Apply \( y(0) = 1 \): By substituting \( x = 0 \) into the general solution \( y(x) = c_1 \cos(3x) + c_2 \sin(3x) \), we get \( 1 = c_1 \cos(0) + c_2 \sin(0) \). This simplifies to \( c_1 = 1 \).
• Apply \( y'(0) = 1 \): First, differentiate the general solution to get \( y'(x) = -3c_1 \sin(3x) + 3c_2 \cos(3x) \). Substituting \( x = 0 \), we get \( 1 = -3c_1 \sin(0) + 3c_2 \cos(0) \). Simplifying this gives \( c_2 = \frac{1}{3} \).

The general solution of a differential equation is a formula that describes all possible solutions. It's expressed in terms of arbitrary constants that later get specified by initial or boundary conditions. For the given differential equation \( y'' + 9y = 0 \), the characteristic equation \( m^2 + 9 = 0 \) has complex roots \( m = \pm 3i \).

These complex roots suggest that the general solution involves sinusoidal functions. In particular, the solution can be written as a combination of sine and cosine functions:

• \( y(x) = c_1 \cos(3x) + c_2 \sin(3x) \).

This form arises because expressions involving \( e^{ix} \) (from Euler's formula) translate to trigonometric functions when roots are imaginary or complex. The constants \( c_1 \) and \( c_2 \) are determined by applying the initial conditions to tailor the solution to the specific problem.

Complex roots in the context of differential equations lead to solutions that are oscillatory in nature. When solving a characteristic equation of the form \( m^2 + b = 0 \), if \( b \) leads to negative under-square-root values, then the roots are complex. For our problem, the characteristic equation \( m^2 + 9 = 0 \) gives complex roots \( m = \pm 3i \).

These complex roots lead the general solution to involve sine and cosine, specifically:

• \( y(x) = c_1 \cos(3x) + c_2 \sin(3x) \).

The coefficients \( c_1 \) and \( c_2 \) become crucial in shaping the trigonometric wave patterns of these solutions. Complex roots reflect the periodic nature often found in physical systems with oscillations like springs, pendulums, or circuits, where solutions must describe sinusoidal behavior. The imaginary unit \( i \) thus transforms our perspective from exponential to trigonometric solutions.