91Ó°ÊÓ

Differential equations are mathematical equations that involve derivatives of a function. They help describe various phenomena where the rate of change is crucial, such as physics, engineering, and other scientific fields. In our problem, we are dealing with a second-order differential equation, which includes the second derivative of the function \( y \) with respect to \( x \). This equation is \( \frac{d^{2} y}{d x^{2}} + y = \sec x \).
This equation is considered non-homogeneous because it is not equal to zero. The presence of the \( \sec x \) term makes it non-homogeneous, as opposed to homogeneous equations where there would be no term on the right-hand side. The goal here is to find a particular solution for this non-homogeneous equation.

A homogeneous equation is a type of differential equation where all its terms are homogeneous functions with the same degree. For our task, the homogeneous version of the original equation is \( \frac{d^{2} y}{d x^{2}} + y = 0 \).
This means there are no external functions like \( \sec x \) affecting it. We use the characteristic equation to solve it, assuming solutions of the form \( y = e^{rx} \).
Solving gives us the roots \( r = i \) and \( r = -i \), which are imaginary. This leads to a general homogeneous solution \( y_h = C_1 \cos x + C_2 \sin x \), utilizing Euler's formula for representing complex exponentials.

In the realm of differential equations, a particular solution refers to a solution that satisfies the non-homogeneous equation. To find it, we use a method known as variation of parameters.
For our problem, given that we already have the complementary solution \( y_h = C_1 \cos x + C_2 \sin x \), we assume a particular solution of the form \( y_p = u_1(x) \cos x + u_2(x) \sin x \), where \( u_1(x) \) and \( u_2(x) \) are unknown functions.
By determining these functions using integration, we find \( u_1(x) = \ln |\cos x| \) and \( u_2(x) = x \). These lead us to the particular solution \( y_p = (\ln |\cos x|) \cos x + x \sin x \). This fits the non-homogeneous differential equation by catering for the additional \( \sec x \) term.

Characteristic roots are crucial in solving homogeneous differential equations. These roots are derived from the characteristic equation, formed by assuming a potential solution of the form \( y = e^{rx} \).
For the homogeneous equation \( \frac{d^{2} y}{d x^{2}} + y = 0 \), its characteristic equation is \( r^2 + 1 = 0 \). Solving it gives us the roots \( r = i \) and \( r = -i \). The roots being imaginary indicates sinusoidal components in the solution.
Hence, the general solution is expressed as \( y_h = C_1 \cos x + C_2 \sin x \), which is a linear combination of sine and cosine functions. These roots help understand the oscillatory nature of the system described by the differential equation.