91Ó°ÊÓ

The characteristic equation is a cornerstone concept when dealing with linear differential equations. It arises from transforming a differential equation by assuming a solution in exponential form. In our original problem, the differential equation is \( y'' + y = 0 \). This is a second-order linear homogeneous differential equation with constant coefficients. To construct the characteristic equation, we begin by presuming a solution of the form \( y = e^{rt} \). By substituting this assumed solution into the differential equation, it leads us to an algebraic equation: the characteristic equation.

• For \( y'' + y = 0 \), substitute \( y = e^{rt} \) into the equation.
• This yields \( r^2 + 1 = 0 \) when simplified.

Hence, the characteristic equation here is \( r^2 + 1 = 0 \). Solving this will guide us toward the form of the general solution.

Solving the characteristic equation \( r^2 + 1 = 0 \) reveals the nature of the roots, which in this case are complex. The equation can be rewritten as \( r^2 = -1 \). The solutions are \( r = \pm i \), indicating complex roots.Complex roots typically come in conjugate pairs, \( a \pm bi \). Despite the apparent complexity, they provide a straightforward path to the general solution. With our roots \( r = \pm i \), the format of the solution, according to Euler's formula, includes cosine and sine functions.

• Complex roots \( a \pm bi \) correspond to solutions \( e^{at}(C_1 \cos(bt) + C_2 \sin(bt)) \).
• Our specific roots \( 0 \pm i \) lead to solutions of the form \( C_1 \cos(t) + C_2 \sin(t) \).

These trigonometric functions are essential as they encapsulate the behavior dictated by the complex roots.

The general solution to a linear differential equation represents the most inclusive form of solutions before specific conditions are applied. Derived from the characteristic roots, our general solution here is \( y(t) = C_1 \cos(t) + C_2 \sin(t) \).The coefficients \( C_1 \) and \( C_2 \) are arbitrary constants that we determine using initial conditions. In this case:

• The cosine coefficient corresponds to the real portion contributed by a complex root.
• The sine coefficient captures the imaginary nature of the solution.

Finding these constants provides the pathway from the general to specific solutions, allowing the model to fit particular initial requirements.

Initial conditions are specific conditions given at the start of a problem to pin down particular solutions from the general solution larger set. In this problem, they are defined as \( y(\pi) = 1 \) and \( y'(\pi) = -5 \).Substituting \( y(\pi) = 1 \) into the general solution \( y(t) = C_1 \cos(t) + C_2 \sin(t) \) lets us express:

• \( C_1 \cos(\pi) + C_2 \sin(\pi) = 1 \)
• This simplifies to \(-C_1 = 1\) since \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \).

Similarly, for \( y'(t) \), which is the derivative \(-C_1 \sin(t) + C_2 \cos(t) \):

• Plugging in \( y'(\pi) = -5 \), we get \(-C_2 = -5\).
• Solving gives \( C_2 = 5 \).

These computations allow us to refine our general solution into a specific solution: \( y(t) = -\cos(t) + 5\sin(t) \), satisfying the initial conditions.