91Ó°ÊÓ

A differential equation is a mathematical equation that relates a function to its derivatives. Solving a differential equation means finding the function that satisfies this relationship. In this exercise, we are dealing with a second-order differential equation. This simply means that the highest derivative in the equation is a second derivative, represented by \( y'' \). The equation given is: \( y'' + y' - 2y = 0 \).
This is an example of a linear homogeneous differential equation. To solve it, we need to find functions \( y \) that, when substituted in place of \( y \), \( y' \), and \( y'' \), satisfy the equation completely, resulting in an identity like \( 0 = 0 \).
In this exercise, we are asked to verify if certain functions, such as \( e^{-2x} \) and \( e^{x} \), as well as their linear combination \( c_1 e^{-2x} + c_2 e^{x} \), are valid solutions. The approach involves calculating the derivatives of these functions and substituting them back in the differential equation to check validity.

Verification is the process of ensuring that a given function truly satisfies the differential equation. Even if a function appears correct at first glance, we can only confirm its status as a solution through verification.
Following this exercise's steps, to verify a function, compute its first and second derivatives. For example, with \( y = e^{-2x} \), the first derivative is \( -2e^{-2x} \), and the second is \( 4e^{-2x} \). After substitution back into the differential equation:

• Substitute \( y'' = 4e^{-2x} \), \( y' = -2e^{-2x} \), and \( y = e^{-2x} \) into \( y'' + y' - 2y = 0 \).
• Each term combines to result in \( 0 = 0 \).

By reaching this equality, we can confirm that the function is indeed a valid solution. This process confirms that each presented function satisfies the equation according to mathematical rules, including the combination \( c_1 e^{-2x} + c_2 e^{x} \), where constants \( c_1 \) and \( c_2 \) are arbitrary.

A second-order differential equation, like the one given in this exercise, is characterized by the presence of a second derivative, \( y'' \). Such equations can describe more complex behaviors in systems, such as oscillations and exponential growth or decay. In general, a second-order differential equation has the form:

• \( a y'' + b y' + c y = 0 \)

where \( a, b, \) and \( c \) are constants. The simplicity or complexity of the solution can vary based on these constants and the initial conditions, if given. Homogeneous second-order equations are particularly significant because they often appear in physics and engineering, modeling behaviors where there's a balance of forces.
For example, the equation \( y'' + y' - 2y = 0 \), which we worked on, is a second-order linear homogeneous differential equation. Solutions often take the form of exponentials like \( e^{mx} \), where \( m \) is derived from the characteristic equation associated with the differential equation. Learning to solve such equations equips one with skills for analyzing real-world systems.